http://www.map.mpim-bonn.mpg.de/api.php?action=feedcontributions&user=Kuessner&feedformat=atomManifold Atlas - User contributions [en]2022-11-27T03:04:00ZUser contributionsMediaWiki 1.18.4http://www.map.mpim-bonn.mpg.de/Talk:Knotted_toriTalk:Knotted tori2017-05-25T00:49:44Z<p>Kuessner: Created page with "Usually one calls a higher-dimensional Torus the product $S^1\times S^1\times\ldots\times S^1$. So it is a bit odd that in embedding theory a knotted torus means $S^p\times S^q$."</p>
<hr />
<div>Usually one calls a higher-dimensional Torus the product $S^1\times S^1\times\ldots\times S^1$. So it is a bit odd that in embedding theory a knotted torus means $S^p\times S^q$.</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Grassmann_manifoldsGrassmann manifolds2013-10-12T17:05:43Z<p>Kuessner: /* Infinite dimensional Grassmannians */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. They play a key role in topology and geometry as the universal spaces of vector bundles. See also [[Wikipedia:Grassmannian|Grassmannian]]<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
</wikitex><br />
<br />
=== Construction ===<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
Let $\F=\Rr ,\Cc , \Hh$ be the real, complex or quaternion field and $V$ a vector space over $\F$ of dimension $n$ and let $k\leq n$. A Grassmannian of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology. Grassmannian is a homogeneous space of the general linear group. General linear group $\GL(V)$ acts transitively on $G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)$ acts transitively and the isotropy group of $W$ is $O(W^\bot)\times O(W)$.<br />
<br />
{{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a Hausdorff, compact, connected smooth manifold of dimension $dk(n-k)$. For $\F=\Cc ,\Hh$ it is also a complex manifold.{{endthm}}<br />
<br />
Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$<br />
<br />
{{beginthm|Proposition|}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}}<br />
</wikitex><br />
=== The canonical bundle ===<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
The Grassmann manifold is equipped with the canonical, tautological vector bundle $\gamma^V_k.$ which is a subbundle of the trivial bundle $G_k(V)\times V \to G_k(V)\times V$. The total space is $E(\gamma^V_k) = \{(W,w)\in G_k(V)\times V\, |\,\, w\in W \}$ The total space of the associated principal bundle is a Stiefel manifold.<br />
<br />
The tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$</wikitex><br />
=== Low dimensional Grassmannians ===<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
The Grassmannians $G_1(V)$ are projective spaces, denoted $P (V)$. Note that $G_1(F^2)=S^d$, where $d=dim_{\Rr} F$. If we identify $S^d$ with the one-point compactification of $\F$ the projection of the canonical principal bundle corresponds to the map $p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$. Note, that the same formula works for octonions $\Oo$, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$ for $d= 2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fibre of $p_d$ is a sphere $S^{d-1}$. <br />
</wikitex><br />
=== Embeddings of Grassmannians into affine and projective space ===<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
There is an embedding of the Grassmannian $G_k(V)$ in the Cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subspace the orthogonal projection on it. If $V$ is equipped with a norm, the embedding defines a natural (operator) metric on $G_k(V)$.<br />
</wikitex><br />
=== Infinite dimensional Grassmannians ===<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
Infinite dimensional Grassmannians. Natural inclusions of vector space $\F ^1 \subset \F ^2 \subset ...\F ^n \subset ...$ defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted $G_k(\F^{\infty} )$ and also $BGL(k,\F)$. One can also take the colimit with respect to both dimension of the space and of the subspaces. We have a sequence of inclusions $G_1(\F^2)\subset G_2(\F^4)\subset ... \subset G_n(\F^{2n}) \subset ...$ and its colimit is denoted $B\GL (\F).$<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
<br />
</wikitex><br />
=== Homotopy groups ===<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
Homotopy groups of Grassmannians are closely related to homotopy groups of spheres via the appropriate fibration sequences. They also imply that the groups $\pi_i(G_k (V))$ do not depend on $V$, if $k\leq\leq dim V.$ Homotopy groups in the stable range are described by the Bott periodicity theorem. <br />
<br />
{{beginthm|Proposition|{(R.Bott)}}} For each $i>0$ there are isomorphisms: $\pi_i(BGL(\Rr)) \simeq \pi_{i+8}(BGL(\Rr))$ and $\pi_i(BGL(\Cc)) \simeq \pi_{i+2}(BGL(\Cc))$ <br />
{{endthm}}<br />
</wikitex><br />
<br />
=== Cohomology groups ===<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
...<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex include="TeXInclude:Grassmann_manifolds">;<br />
Grassmann manifolds are examples of [[coadjoint orbits]] \cite{Kirillov2004}.<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
==External links==<br />
* The Wikipedia page on [[Wikipedia:Grassmannian|Grassmannian]]<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Talk:1-manifoldsTalk:1-manifolds2013-04-17T05:57:56Z<p>Kuessner: Created page with "There is also a vaste theory for infinite group actions on the circle, see http://www.math.ethz.ch/~bgabi/ghys%20groups%20acting%20on%20the%20circle.pdf Moreover, in the conte..."</p>
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<div>There is also a vaste theory for infinite group actions on the circle, see http://www.math.ethz.ch/~bgabi/ghys%20groups%20acting%20on%20the%20circle.pdf<br />
Moreover, in the context of foliation theory people studied group actions on non-Hausdorff 1-manifolds.</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Linking_formLinking form2013-03-31T06:24:40Z<p>Kuessner: </p>
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<div>{{Stub}}<br />
==Background: intersection forms==<br />
<wikitex>;<br />
After Poincaré and Lefschetz, a closed oriented manifold $N^{n}$ has a bilinear [[Intersection form|intersection form]] defined on its homology. Given a ${k}$--chain $p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$--chain $q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$, the signed count of the intersections between $p$ and $q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$.<br />
<br />
<br />
The intersection form is defined by<br />
$$I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$$<br />
and is such that<br />
$$I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$$<br />
</wikitex><br />
<br />
== Definition of the linking form==<br />
<wikitex>;<br />
By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$ is the subgroup $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$$<br />
<br />
The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$--valued linking form, which is due to Seifert \cite{Seifert1933}:<br />
$$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$<br />
such that<br />
$$L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$$<br />
and computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{n-\ell}(N;\mathbb{Z})$ be such that $\partial w = sy$, for some $s \in \mathbb{Z}$. Then we define:<br />
$$L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$$<br />
The resulting element is independent of the choices of $x,y,w$ and $s$. <br />
</wikitex><br />
<br />
==Definition via cohomology==<br />
<wikitex>;<br />
Let $x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré duality isomorphisms<br />
$$PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})$$<br />
and<br />
$$PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).$$<br />
Associated to the short exact sequence of coefficients<br />
$$0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$$<br />
is the Bockstein long exact sequence in cohomology.<br />
$$H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell}(N;\mathbb{Q}).$$<br />
Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$. There is a cup product:<br />
$$\cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$$<br />
Compute $a:= z \cup PD(y)$. Then the Kronecker pairing:<br />
$$\langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}$$<br />
of $a$ with the fundamental class of $N$ yields $L_N(x,y)$.<br />
</wikitex><br />
<br />
==Example of 3-dimensional projective space==<br />
<wikitex>;<br />
As an example, let $N = \mathbb{RP}^3$, so that $\ell=1$ and $n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non-trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$ modelled as $D^3/\sim$, with antipodal points on $\partial D^2$ identified, and choose two representative $1$-chains $x$ and $y$ for $\theta$. Let $x$ be the straight line between north and south poles and let $y$ be half of the equator. Now $2y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$ is the 2-disk whose boundary is the equator. We see that $\langle x,w \rangle = 1$, so that<br />
$$L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$$<br />
</wikitex><br />
==Example of lens spaces==<br />
<wikitex>;<br />
Generalising the above example, the 3-dimensional lens space $N_{p,q}$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$. The linking form is given on a generator $\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = q/p$. Note that $L_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example.<br />
</wikitex><br />
<br />
==Presentations of linking forms==<br />
<wikitex>;<br />
A presentation for a middle dimensional linking form on $N^{2\ell +1}$<br />
$$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{\ell}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$<br />
is an exact sequence:<br />
<br />
$$0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$<br />
<br />
where $F$ is a free abelain group and the linking $L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$ be such that $\partial(x')=x$ and $\partial(y')=y$. Then we can tensor with $\mathbb{Q}$ to obtain an isomorphism <br />
$$\Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$$<br />
The linking form is given by:<br />
$$L_N(x,y) = (x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$$<br />
<br />
<br />
Let $\ell = 1$, so $2\ell + 1 = 3$. Every 3-manifold $N$ is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in $S^3$ \cite{Lickorish1962}, \cite{Wallace1960}. This is sometimes called a surgery presentation for $N$. Suppose that $N$ is a rational homology 3-sphere. Let $A$ be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for $N$ as the rank of $F$, the linking matrix $A$ determines a map $\Phi$ as above, which presents the linking form of $N$. The intersection form on a simply connected 4-manifold $W$ whose boundary is $N$ presents the linking form of $N$. This follows from the long exact sequence of the pair~$(W,N)$ and Poincar\'{e} duality. See \cite{Boyer1986} for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.<br />
</wikitex><br />
<br />
==Classification of 5-manifolds==<br />
<br />
Linking forms are central to the classification of simply connected 5-manifolds. See this [[5-manifolds: 1-connected|5-manifolds]] page, which also describes the classification of anti-symmetric linking forms.<br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Definitions]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Linking_formLinking form2013-03-31T06:23:41Z<p>Kuessner: /* Definition of the linking form */</p>
<hr />
<div>{{Stub}}<br />
==Background: intersection forms==<br />
<wikitex>;<br />
After Poincaré and Lefschetz, a closed oriented manifold $N^{n}$ has a bilinear [[Intersection form|intersection form]] defined on its homology. Given a ${k}$--chain $p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$--chain $q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$, the signed count of the intersections between $p$ and $q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$.<br />
<br />
<br />
The intersection form is defined by<br />
$$I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$$<br />
and is such that<br />
$$I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$$<br />
</wikitex><br />
<br />
== Definition of the linking form==<br />
<wikitex>;<br />
By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$ is the subgroup $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$$<br />
<br />
The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$--valued linking form, which is due to Seifert \cite{Seifert1933}:<br />
$$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$<br />
such that<br />
$$L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$$<br />
and computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{n-\ell}(N;\mathbb{Z})$ be such that $\partial w = sy$, for some $s \in \mathbb{Z}$. Then we define:<br />
$$L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$$<br />
The resulting element is independent of the choices of $x,y,w$ and $s$. <br />
</wikitex><br />
<br />
==Definition via homology==<br />
<wikitex>;<br />
Let $x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré duality isomorphisms<br />
$$PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})$$<br />
and<br />
$$PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).$$<br />
Associated to the short exact sequence of coefficients<br />
$$0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$$<br />
is the Bockstein long exact sequence in cohomology.<br />
$$H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell}(N;\mathbb{Q}).$$<br />
Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$. There is a cup product:<br />
$$\cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$$<br />
Compute $a:= z \cup PD(y)$. Then the Kronecker pairing:<br />
$$\langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}$$<br />
of $a$ with the fundamental class of $N$ yields $L_N(x,y)$.<br />
</wikitex><br />
<br />
==Example of 3-dimensional projective space==<br />
<wikitex>;<br />
As an example, let $N = \mathbb{RP}^3$, so that $\ell=1$ and $n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non-trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$ modelled as $D^3/\sim$, with antipodal points on $\partial D^2$ identified, and choose two representative $1$-chains $x$ and $y$ for $\theta$. Let $x$ be the straight line between north and south poles and let $y$ be half of the equator. Now $2y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$ is the 2-disk whose boundary is the equator. We see that $\langle x,w \rangle = 1$, so that<br />
$$L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$$<br />
</wikitex><br />
==Example of lens spaces==<br />
<wikitex>;<br />
Generalising the above example, the 3-dimensional lens space $N_{p,q}$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$. The linking form is given on a generator $\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = q/p$. Note that $L_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example.<br />
</wikitex><br />
<br />
==Presentations of linking forms==<br />
<wikitex>;<br />
A presentation for a middle dimensional linking form on $N^{2\ell +1}$<br />
$$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{\ell}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$<br />
is an exact sequence:<br />
<br />
$$0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$<br />
<br />
where $F$ is a free abelain group and the linking $L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$ be such that $\partial(x')=x$ and $\partial(y')=y$. Then we can tensor with $\mathbb{Q}$ to obtain an isomorphism <br />
$$\Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$$<br />
The linking form is given by:<br />
$$L_N(x,y) = (x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$$<br />
<br />
<br />
Let $\ell = 1$, so $2\ell + 1 = 3$. Every 3-manifold $N$ is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in $S^3$ \cite{Lickorish1962}, \cite{Wallace1960}. This is sometimes called a surgery presentation for $N$. Suppose that $N$ is a rational homology 3-sphere. Let $A$ be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for $N$ as the rank of $F$, the linking matrix $A$ determines a map $\Phi$ as above, which presents the linking form of $N$. The intersection form on a simply connected 4-manifold $W$ whose boundary is $N$ presents the linking form of $N$. This follows from the long exact sequence of the pair~$(W,N)$ and Poincar\'{e} duality. See \cite{Boyer1986} for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.<br />
</wikitex><br />
<br />
==Classification of 5-manifolds==<br />
<br />
Linking forms are central to the classification of simply connected 5-manifolds. See this [[5-manifolds: 1-connected|5-manifolds]] page, which also describes the classification of anti-symmetric linking forms.<br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Definitions]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationFoliation2013-03-27T12:15:35Z<p>Kuessner: /* Foliations */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
This page gives the definition of the term ''foliation''. For further information, see the page [[Foliations]].<br />
=== Foliations ===<br />
<wikitex>;<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
{{cite|Godbillon1991}}<br />
</wikitex><br />
<br />
=== Defining differential form ===<br />
<wikitex>;<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
</wikitex><br />
=== Leaves ===<br />
<wikitex>;<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
</wikitex><br />
=== Holonomy Cocycle ===<br />
<wikitex>;<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
</wikitex><br />
<br />
== Special classes of foliations ==<br />
=== Bundles ===<br />
<wikitex>;<br />
The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)<br />
<br />
A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)<br />
<br />
Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a homeomorphism (diffeomorphism) $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:<br />
$$\begin{xy}<br />
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\<br />
U\ar[r]^{id}&U}<br />
\end{xy}$$<br />
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.<br />
</wikitex><br />
<br />
=== Suspensions ===<br />
<wikitex>;<br />
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=<br />
\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$$<br />
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.<br />
<br />
In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.<br />
<br />
The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. <br />
If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. <br />
</wikitex><br />
<br />
=== Submersions ===<br />
<wikitex>;<br />
Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. <br />
[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]<br />
An example of a submersion, which is not a fiber bundle, is given by<br />
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$<br />
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$<br />
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.<br />
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$<br />
for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the Möbius band). Their leaf spaces are not Hausdorff.<br />
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]<br />
</wikitex><br />
<br />
=== Reeb foliations ===<br />
<wikitex>;<br />
Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by<br />
$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$<br />
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ <br />
for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.<br />
</wikitex><br />
<br />
=== Taut foliations ===<br />
<wikitex>;<br />
A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. <br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
== External links ==<br />
* The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Foliation foliations]<br />
* The Wikipedia page about [[Wikipedia:Foliation|foliations]]<br />
<br />
[[Category:Definitions]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationFoliation2013-03-27T10:26:04Z<p>Kuessner: </p>
<hr />
<div><br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
=== Foliations ===<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
<br />
=== Defining differential form ===<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
<br />
=== Leaves ===<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
<br />
=== Holonomy Cocycle ===<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
<br />
<br />
== Special classes of foliations ==<br />
=== Bundles ===<br />
<br />
<wikitex>;<br />
<br />
The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)<br />
<br />
A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)<br />
<br />
Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a homeomorphism (diffeomorphism) $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:<br />
$$\begin{xy}<br />
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\<br />
U\ar[r]^{id}&U}<br />
\end{xy}$$<br />
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.<br />
</wikitex><br />
<br />
=== Suspensions ===<br />
<wikitex>;<br />
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=<br />
\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$$<br />
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.<br />
<br />
In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.<br />
<br />
The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. <br />
If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. <br />
</wikitex><br />
<br />
=== Submersions ===<br />
<wikitex>;<br />
Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. <br />
[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]<br />
An example of a submersion, which is not a fiber bundle, is given by<br />
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$<br />
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$<br />
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.<br />
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$<br />
for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the Möbius band). Their leaf spaces are not Hausdorff.<br />
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]<br />
</wikitex><br />
<br />
=== Reeb foliations ===<br />
<wikitex>;<br />
Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by<br />
$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$<br />
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ <br />
for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.<br />
<br />
</wikitex><br />
<br />
=== Taut foliations ===<br />
<wikitex>;<br />
A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. <br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Definitions]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationFoliation2013-03-27T10:23:25Z<p>Kuessner: </p>
<hr />
<div><br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
=== Foliations ===<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
<br />
=== Defining differential form ===<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
<br />
=== Leaves ===<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
<br />
=== Holonomy Cocycle ===<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
<br />
<br />
== Special classes of foliations ==<br />
=== Bundles ===<br />
<br />
<wikitex>;<br />
<br />
The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)<br />
<br />
A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)<br />
<br />
Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a homeomorphism (diffeomorphism) $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:<br />
$$\begin{xy}<br />
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\<br />
U\ar[r]^{id}&U}<br />
\end{xy}$$<br />
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.<br />
</wikitex><br />
<br />
=== Suspensions ===<br />
<wikitex>;<br />
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=<br />
\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$$<br />
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.<br />
<br />
In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.<br />
<br />
The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. <br />
If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. <br />
</wikitex><br />
<br />
=== Submersions ===<br />
<wikitex>;<br />
Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. <br />
[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]<br />
An example of a submersion, which is not a fiber bundle, is given by<br />
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$<br />
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$<br />
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.<br />
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$<br />
for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the Möbius band). Their leaf spaces are not Hausdorff.<br />
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]<br />
</wikitex><br />
<br />
=== Reeb foliations ===<br />
<wikitex>;<br />
Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by<br />
$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$<br />
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ <br />
for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.<br />
<br />
</wikitex><br />
<br />
=== Taut foliations ===<br />
<wikitex>;<br />
A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. <br />
<br />
{{beginthm|Theorem|(Rummler, Sullivan)}}<br />
The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$:<br />
<br />
a) $\mathcal{F}$ is taut;<br />
<br />
b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$;<br />
<br />
c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}} <br />
<br />
<br />
</wikitex><br />
<br />
== Constructing new foliations from old ones ==<br />
<br />
<br />
==== Pullbacks ====<br />
<wikitex>;<br />
{{beginthm|Theorem |}} If $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ and $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$, then $N$ is foliated by connected components of $f^{-1}\left(L\right)$ as $L$ ranges over the leaves of $\mathcal{F}$. {{endthm}}<br />
{{cite|Candel&Conlon2000}}, Theorem 3.2.2<br />
</wikitex><br />
<br />
==== Glueing ====<br />
<wikitex>;<br />
Let $\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$ be $n$-manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$.<br />
If either both foliations are tangent or both foliations are transverse to the boundaries of $M_1$ and $M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$. This is called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$.<br />
</wikitex><br />
<br />
==== Turbulization ====<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$. <br />
<br />
Define a foliation ${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by<br />
$$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$<br />
where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$<br />
The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhood of the boundary of $N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$ of $\mathcal{F}$.<br />
</wikitex><br />
<br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Definitions]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationFoliation2013-03-27T10:20:53Z<p>Kuessner: Created page with " {{Stub}} == Introduction == <wikitex>; == Introduction == <wikitex>; === Foliations === Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\..."</p>
<hr />
<div><br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
== Introduction ==<br />
<wikitex>;<br />
=== Foliations ===<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
<br />
=== Defining differential form ===<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
<br />
=== Leaves ===<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
<br />
=== Holonomy Cocycle ===<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
<br />
<br />
</wikitex><br />
</wikitex><br />
== Definition ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
== Examples ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Definitions]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Connected_sumConnected sum2013-03-02T15:50:47Z<p>Kuessner: /* Examples */</p>
<hr />
<div>{{Stub}}<br />
== Connected sum of smooth manifolds ==<br />
<wikitex>;<br />
Let $M_0$ and $M_1$ be oriented closed smooth connected $n$-manifolds. Their connected sum is an oriented closed smooth connected $n$-manifold<br />
$$ M_1 \sharp M_2 $$<br />
which is defined as follows (c.f. \cite{Kervaire&Milnor1963|Section 2}). Choose smooth embeddings <br />
$$ i_0 \colon D^n \to M_0 \quad \text{and} \quad i_1 \colon D^n \to M_1 $$<br />
where $i_1$ preserves orientations and $i_2$ reverses orientations. The connected sum is formed from the<br />
disjoint union<br />
$$ \bigl( M_0 - i_0(0) \bigr) \sqcup \bigl(M_1 - i_1(0) \bigr) $$<br />
by identifying $i_0(tu)$ with $i_1((1-t)u)$ for $u \in S^{n-1}$ and $0 < t < 1$. The smooth structure on <br />
$M_0 \sharp M_1$ is obtain from the charts on $M_0 - i_0(0)$ and $M_1 - i_1(0)$. The orientation on <br />
$M_0 \sharp M_1$ is chosen to be the one compatible with the orientation of $M_0$ and $M_1$.<br />
<br />
A fundamental lemma of differential topology, \cite{Palais1959|Theorem 5.5} \cite{Cerf1961} states that any<br />
two orientation preserving smootgh embeddings of the $n$-disc into a closed oriented smooth $n$-manifold are isotopic. As a consequence we have the following lemma.<br />
<br />
{{beginthm|Lemma|\cite{Kervaire&Milnor1963|Lemma 2.1} }}<br />
The connected sum operation is well defined, associative and commutative up to <br />
orientation preserving diffeomoprhism. The sphere $S^n$ serves as the identity element.<br />
{{endthm}}<br />
<br />
The connected sum operation also descends to give well-defined operations on larger equivalence classes<br />
of oriented manifolds.<br />
<br />
{{beginthm|Lemma|c.f. \cite{Kervaire&Milnor1963|Lemma 2.2} }}<br />
Let $M_0$, $M_0'$ and $M_1$ be oriented closed connected smooth manifold. Suppose that $M_0$ is [[h-cobordism|h-cobordant]] to $M_0'$, resp. bordant to $M_0'$ then $M_0 \sharp M_1$ is h-cobordant, resp. bordant, to $M_0' \sharp M_1$.<br />
{{endthm}}<br />
</wikitex><br />
<!-- If $M$ and $N$ are oriented manifolds the connected sum $M \sharp N$ is a well-defined up to diffeomorphism. Note that orientation matters! --><br />
<!-- Let $M$ be a compact connected n-manifold with base point $m \in \mathrm{int} M$. Recall that that a local orientation for $M$ is a choice of orientation of $TM_m$, the tangent space to $M$ at $m$. We write $-M$ for $M$ with the opposition orientation at $m$. Of course, if $M$ is orientable then a local orientation for $M$ defines an orientation on $M$.<br />
If $M$ and $N$ are locally oriented n-manifolds then their [[Wikipedia:Connected_sum|connected sum]] is defined by <br />
$$ M \sharp N = ((M - m) \cup (N - n))/ \simeq$$ <br />
where $\simeq$ is defined using the local orientations to identify small balls about $m$ and $n$. The diffeomorphism type of $M \sharp N$ is well-defined: in fact $M \sharp N$ is the outcome of 0-surgery on $M \sqcup N$. The essential point is \cite{Hirsch} which states, for any $M$ and any two compatibly oriented embeddings $f_1: D^n \to M$ and $f_1 : D^n \to M$, that $f_0$ is isotopic to $f_1$. --><br />
<br />
== Connected sum of topological manifolds ==<br />
<wikitex>;<br />
Connected sum is a well-defined operation up to orientation preserving homeomorphism for oriented closed connected topological $n$-manifolds. However, there is no analogue of the Palais/Cerf result and so the proof is more complicated. See the mathoverflow discussion cited [[#External links|below]].<br />
</wikitex><br />
<br />
== Examples==<br />
<wikitex>;<br />
The orientation of the manifolds is important in general. The canonical example is<br />
$$ \CP^2 \sharp \CP^2 \neq \CP^2 \sharp (-\CP^2).$$<br />
The manifolds are not even homotopy equivalent: the first has signature 2 the other signature 0. The following elementary lemma is often useful to remember.<br />
{{beginthm|Lemma}}<br />
Let $M$ and $N$ be locally oriented manifolds such that there is a diffeomoprhism $N \cong -N$, then $M \sharp N \cong M \sharp (-N)$.<br />
{{endthm}}<br />
<br />
Connected sum decompositions of manifolds are far from being unique. For example, let $M = S^3 \tilde \times S^4$ be the total space of the non-trivial 3-sphere bundle over $S^4$ with Euler class zero and Pontrjagin class four times a preferred generator of $H^4(S^4; \Z) \cong \Z$. <br />
{{beginthm|Lemma|c.f.\cite{Wilkens1974/75|Theorem 1} }}<br />
There are diffeomorphisms<br />
# $M \sharp M \cong M \sharp (S^3 \times S^4)$<br />
# $M \sharp \Sigma \cong M$ for any homotopy sphere $\Sigma$. (Recall that the group of [[Exotic spheres|homotopy 7-spheres]], $\Theta_7$ is isomorphic to $\Z/28$.)<br />
{{endthm}}<br />
{{beginproof}}<br />
1.) The manifold $M$ is the boundary of the total space of the corresponding disc bundle $W : = D^4 \tilde \times S^4$ and hence $M \sharp M$ is the boundary of $W \natural W$. Compact $3$-connected $8$-manifolds were classified<br />
in \cite{Wall1962a|Section 2}. Since the intersection form of $W \natural W$ is trivial, it is a simple consequence<br />
of Wall's classification that there is a diffeomorphism $f \colon W \natural W \cong W \natural (D^4 \times S^4)$. Restricting $f$ to the boundary gives the desired diffeomorphism.<br />
<br />
2.) This is a special case of \cite{Wilkens1974/75|Theorem 1}.<br />
{{endproof}}<br />
</wikitex><br />
<br />
== Properties ==<br />
<wikitex>;<br />
Let $M$ be a closed connected $n$-manifold and let <br />
$$M^\bullet : = M \setminus \textup{Int}(D^n)$$ <br />
denote the compact manifold obtained from $M$ by deleting a small embedded open $n$-disc. From the definition<br />
it is clear that<br />
$$ (M_0 \sharp M_1)^\bullet = (M_0^\bullet) \natural (M_1^\bullet) \simeq M_0^\bullet \vee M_1^\bullet.$$<br />
Here $\vee$ denotes the one point union of topological spaces and $\simeq$ indicates that two spaces are<br />
homotopy equivalent. Applying Van Kampen's Theorem we immediately obtain the following lemma.<br />
{{beginthm|Lemma}}<br />
Let the dimension $n$ be three or greater. Then the fundamental group of a connected sum is the free <br />
product of the fundamental group of the components:<br />
$$ \pi_1(M_0 \sharp M_1) \cong \pi_1(M_0) \ast \pi_1(M_2) .$$<br />
{{endthm}}<br />
</wikitex><br />
== References ==<br />
{{#RefList:}}<br />
== External links ==<br />
* [http://mathoverflow.net/questions/121571/connected-sum-of-topological-manifolds Mathoverflow:Connected sum of topological manifolds]<br />
<br />
[[Category:Theory]]<br />
[[Category:Definitions]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hirsch-Smale_theoryHirsch-Smale theory2012-12-07T05:46:20Z<p>Kuessner: /* Applications */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
An immersion $f:A\rightarrow N$ is a map of manifolds which is locally an embedding, i.e. such that for<br />
each $a \in A$ there exists an open neighbourhood $U \subseteq A$ with $a \in U$ and $f\vert:U \to N$ an embedding.<br />
A regular homotopy of immersions $f_0,f_1:A \rightarrow N$ is a homotopy $h:f_0 \simeq f_1:A \rightarrow N$ <br />
such that each $h_t:A \rightarrow N$ ($t \in I$) is an immersion.<br />
<br />
Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the [[h-principle]].<br />
</wikitex><br />
<br />
== Results ==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} For a submanifold $A\subset{\mathbb R}^q$ and a manifold $N$, a pair $\left(f,f^\prime\right)$ is called an ${\mathbb R}^q$-immersion if <br />
<br />
- $f:A\rightarrow N$ is an immersion,<br />
<br />
- $f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$ is a linear bundle map, and<br />
<br />
- there exists an open neighborhood $U$ of $A$ in ${\mathbb R}^q$ and an immersion $g:U\rightarrow N$ such that $g\mid_{A}=f$ and $Dg\mid_{A}=f^\prime$. <br />
{{endthm}}<br />
<br />
{{beginthm|Definition}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$-immersion. The obstruction to extending $\left(f,f^\prime\right)$, denoted by $\tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right)$ with $V_{n,q}$ the Stiefel manifold of $q$-frames in ${\mathbb R}^n$, is the homotopy class of $$x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).$${{endthm}}<br />
<br />
{{beginthm|Theorem |}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$-immersion. <br />
<br />
If $k+1<n$ and $\tau\left(f^\prime\right)=0$, then $\left(f,f^\prime\right)$ can be extended to an ${\mathbb R}^q$-immersion $f:D^{k+1}\rightarrow {\mathbb R}^n$. {{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 3.9.<br />
<br />
This theorem does not hold for $n=k+1$. <br />
<br />
If $n=k+1=2$, then conditions for the extendibility of $\left(f,f^\prime\right)$ are given in {{cite|Blank1967}} (see also \cite{Poénaru1995}), more details are worked out in {{cite|Frisch2010}}.<br />
</wikitex><br />
<br />
==Applications==<br />
<wikitex>;<br />
{{beginthm|Theorem |}}<br />
Let $M$ be a smooth manifold of dimension $k<n$. Then the following assertions are equivalent:<br />
<br />
(i) $M$ can be immersed into ${\mathbb R}^n$,<br />
<br />
(ii) there exists a $GL\left(k,{\mathbb R}\right)$-equivariant map $T_k\left(M\right)\rightarrow V_{n,k}$, where $T_k\left(M\right)\rightarrow M$ is the $k$-frame bundle and $V_{n,k}$ is the Stiefel manifold,<br />
<br />
(iii) the bundle associated to $T_k \left( M \right)$ with fiber $V_{n,k}$ has a cross section.<br />
{{endthm}}<br />
{{cite|Hirsch1959|Theorem 6.1}}<br />
The equivalence between (i) and (ii) is proved by induction over the dimension of subsimplices in a triangulation of $M$ using Theorem 3.9 (which can be adapted from $\left(D^k,S^{k-1}\right)$ to $\left(\Delta^k,\partial \Delta^k\right)$) for the inductive step. The equivalence between (ii) and (iii) is a general fact from the theory of fiber bundles.<br />
{{beginthm|Corollary |}} Parallelizable $k$-manifolds can be immersed into ${\mathbb R}^{k+1}$.{{endthm}}<br />
{{beginthm|Corollary |}} Compact $3$-manifolds can be immersed into ${\mathbb R}^4$.{{endthm}}<br />
{{beginthm|Corollary |}} Exotic $7$-spheres can be immersed into ${\mathbb R}^8$.{{endthm}}<br />
</wikitex><br />
<br />
==References==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationsFoliations2012-09-20T11:41:20Z<p>Kuessner: /* Suspensions */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
=== Bundles ===<br />
<br />
<wikitex>;<br />
<br />
The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)<br />
<br />
A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)<br />
<br />
Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a homeomorphism (diffeomorphism) $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:<br />
$$\begin{xy}<br />
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\<br />
U\ar[r]^{id}&U}<br />
\end{xy}$$<br />
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.<br />
</wikitex><br />
<br />
=== Suspensions ===<br />
<wikitex>;<br />
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=<br />
\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$$<br />
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.<br />
<br />
In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.<br />
<br />
The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. <br />
If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. <br />
</wikitex><br />
<br />
=== Submersions ===<br />
<wikitex>;<br />
Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. <br />
[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]<br />
An example of a submersion, which is not a fiber bundle, is given by<br />
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$<br />
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$<br />
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.<br />
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$<br />
for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the M&ouml;bius band). Their leaf spaces are not Hausdorff.<br />
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]<br />
</wikitex><br />
<br />
=== Reeb foliations ===<br />
<wikitex>;<br />
Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by<br />
$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$<br />
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ <br />
for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.<br />
<br />
</wikitex><br />
<br />
=== Taut foliations ===<br />
<wikitex>;<br />
A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. <br />
<br />
{{beginthm|Theorem|(Rummler, Sullivan)}}<br />
The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$:<br />
<br />
a) $\mathcal{F}$ is taut;<br />
<br />
b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$;<br />
<br />
c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}} <br />
<br />
<br />
</wikitex><br />
<br />
=== Constructing new foliations from old ones ===<br />
<br />
==== Pullbacks ====<br />
<wikitex>;<br />
{{beginthm|Theorem |}} If $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ and $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$, then $N$ is foliated by connected components of $f^{-1}\left(L\right)$ as $L$ ranges over the leaves of $\mathcal{F}$. {{endthm}}<br />
{{cite|Candel&Conlon2000}}, Theorem 3.2.2<br />
</wikitex><br />
<br />
==== Glueing ====<br />
<wikitex>;<br />
Let $\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$ be $n$-manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$.<br />
If either both foliations are tangent or both foliations are transverse to the boundaries of $M_1$ and $M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$. This is called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$.<br />
</wikitex><br />
<br />
==== Turbulization ====<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$. <br />
<br />
Define a foliation ${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by<br />
$$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$<br />
where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$<br />
The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhood of the boundary of $N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$ of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<br />
=== Holonomy ===<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation and $L$ a leaf. For a path $\gamma:\left[0,1\right]\rightarrow L$ contained in the intersection of the leaf $L$ with<br />
a foliation chart $U$, and two transversals $\tau_0,\tau_1$ to $\gamma$ at the endpoints, the product structure of the foliation chart determines a homeomorphism $$h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$$<br />
If $\gamma$ is covered by foliation charts $U_0,\ldots,U_k$, then one obtains a sequence of homeomorphisms $$h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$$<br />
The composition yields a well-defined map $h$ from the germ of $\tau_0$ at $\gamma\left(0\right)$<br />
to the germ of $\tau_{k+1}$ at $\gamma\left(1\right)$, the so-called holonomy transport. The holonomy transport only depends on the relative homotopy class of $\gamma$. <br />
{{beginthm|Lemma}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation, $L$ a leaf, $x\in L$ and $\tau$ a transversal at $x$. Holonomy transport defines a homomorphism $$H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)$$ to the group of germs of homeomorphisms of $\tau$.<br />
{{endthm}}<br />
<br />
{{beginthm|Corollary|(Reeb)}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension one foliation of a 3-manifold such that some leaf $L$ is homeomorphic to $S^2$. Then $M=S^2\times S^1$ and $\mathcal{F}$ is the product foliation by spheres.<br />
{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.5<br />
<br />
<br />
</wikitex><br />
=== Godbillon-Vey invariant ===<br />
<wikitex>;<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation of a manifold $M$, then its tangential plane field $E$ is defined by a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ and $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. The Godbillon-Vey invariant of $\mathcal{F}$ is defined as<br />
$$gv\left({\mathcal{F}}\right):=\left[\eta\wedge\left(d\eta\right)^q\right]\in H^{2q+1}_{dR}\left(M\right).$$<br />
<br />
The Godbillion-Vey invariant is related to resilience of leaves. A leaf is said to be resilient if it is not properly embedded and its holonomy is not trivial.<br />
<br />
{{beginthm|Theorem|(Duminy)}}<br />
If $\left(M,{\mathcal{F}}\right)$ is a foliation of codimension one and no leaf is resilient, then $gv\left({\mathcal{F}}\right)=0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== Classification ==<br />
<br />
=== Codimension one foliations ===<br />
<br />
==== Existence ====<br />
<wikitex>;<br />
{{beginthm|Theorem}}<br />
A closed smooth manifold $M^n$ has a smooth codimension one foliation if and only if $\chi(M^n)=0$, where $\chi$ denotes the Euler characteristic. <br />
<br />
If $\chi(M^n)=0$, then every $(n-1)$-plane field $\tau^{n-1}$ on $M^n$ is homotopic to the tangent plane field of a smooth codimension one foliation.{{endthm}}<br />
<br />
{{cite|Thurston1976}}<br />
</wikitex><br />
<br />
==== Foliations of surfaces ====<br />
<wikitex>;<br />
If $\left({\mathbb R}^2,{\mathcal{F}}\right)$ is a codimension one foliation of the plane ${\mathbb R}^2$, then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold $L$. This provides a 1-1-correspondence between foliations of ${\mathbb R}^2$ and simply connected 1-manifolds. <br />
<br />
Codimension one foliations on compact surfaces $S$ exist only if $\chi\left(S\right)=0$, that is on the Torus, the Klein bottle, the annulus and the Möbius band.<br />
<br />
A foliation $\left(S,{\mathcal{F}}\right)$ is said to contain a Reeb component resp. a non-orientable Reeb component if the restriction of ${\mathcal{F}}$ to some subsurface $S^\prime\subset S$ is a Reeb foliation resp. a non-orientable Reeb foliation. (This implies that $S^\prime$ is an annulus resp. a Möbius band.)<br />
{{beginthm|Theorem}} <br />
<br />
a) Let $\left(S,{\mathcal{F}}\right)$ be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:<br />
<br />
(1) $\mathcal{F}$ is the suspension of a homeomorphism $f:S^1\rightarrow S^1$ or<br />
<br />
(2) $\mathcal{F}$ contains a Reeb component (orientable or not).<br />
<br />
b) Every foliation of the annulus $S^1\times I$ tangent to the boundary is obtained by glueing together a finite number of Reeb components and a finite number of suspensions<br />
<br />
c) Every foliation of the Möbius band tangent to the boundary is one of the following three possibly glued together with a foliation on $S^1\times I$:<br />
<br />
(1) the non-orientable Reeb component<br />
<br />
(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution<br />
<br />
(3) the suspension of an orientation-reversing homeomorpism $f:I\rightarrow I$.{{endthm}}<br />
{{cite|Hector&Hirsch1981}}, Theorem 4.2.15 and Proposition 4.3.2<br />
</wikitex><br />
<br />
==== Foliations of 3-manifolds ====<br />
<wikitex>;<br />
{{beginthm|Theorem|(Novikov)}} If a 3-manifold $M$ admits a foliation $\mathcal{F}$ without Reeb components, then $\pi_2\left(M\right)=0$, every leaf of $\mathcal{F}$ is incompressible, and every transverse loop is essential in $\pi_1\left(M\right)$. {{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.37<br />
<br />
A taut foliation has no Reeb component. If $M$ is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.<br />
<br />
{{beginthm|Theorem|(Palmeira)}} If $\mathcal{F}$ is a taut foliation of a 3-manifold $M$ not finitely covered by $S^2\times S^1$, then the universal covering $\widetilde{M}$ is homeomorphic to ${\mathbb R}^3$ and the pull-back foliation $\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$ is homeomorphic to a product foliation $\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$, where $\mathcal{G}$ is a foliation of ${\mathbb R}^2$ by lines.{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.38<br />
<br />
In particular, a taut foliation of a 3-manifold $M$ yields an action of $\pi_1M$ on a (possibly non-Hausdorff) simply connected 1-manifold $L$, the space of leaves of ${\mathcal{G}}$.<br />
<br />
{{beginthm|Theorem|(Gabai)}} Let $M$ be a closed, irreducible 3-manifold.<br />
<br />
a) If $H_2\left(M;{\mathbb R}\right)\not =0$, then $M$ admits a taut foliation.<br />
<br />
b) If $S$ is a surface which minimizes the Thurston norm in its homology class $\left[S\right]\in H_2\left(M;{\mathbb R}\right)$, then $M$ admits a taut foliaton for which $S$ is a leaf.{{endthm}}<br />
<br />
=== Codimension two foliations ===<br />
<br />
==== $S^1$-foliations of 3-manifolds ====<br />
<br />
{{beginthm|Theorem|(Epstein)}} Every foliation of a compact 3-manifold by circles is a Seifert fibration.{{endthm}}<br />
<br />
{{beginthm|Example}}<br />
a) For every rational number $\frac{p}{q}\not=0$ there exists a foliaton of $S^3=\left\{\left(z,w\right)\in{\mathbb C}^2: \mid z\mid^2+\mid w\mid^2=1\right\}$ by circles such that restriction to the standard embedded torus $\left\{\left(z,w\right)\in S^3: \mid z\mid=\mid w\mid=1\right\}$ is the suspension foliation of $R_{\frac{p}{q}2\pi}$. <br />
b) The complement of a knot $K\subset S^3$ admits a foliation by circles if and only if $K$ is a torus knot.{{endthm}}<br />
<br />
{{beginthm|Theorem|(Vogt)}} If a 3-manifold $M$ admits a foliation by circles, then any 3-manifold obtained by removing finitely many points from $M$ admits a (not necessarily smooth) foliation by circles.{{endthm}}<br />
<br />
{{beginthm|Corollary}} ${\mathbb R}^3$ admits a foliation by circles.{{endthm}}<br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
{{Stub}}</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationsFoliations2012-09-20T11:27:43Z<p>Kuessner: /* Bundles */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
=== Bundles ===<br />
<br />
<wikitex>;<br />
<br />
The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)<br />
<br />
A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)<br />
<br />
Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a homeomorphism (diffeomorphism) $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:<br />
$$\begin{xy}<br />
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\<br />
U\ar[r]^{id}&U}<br />
\end{xy}$$<br />
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.<br />
</wikitex><br />
<br />
=== Suspensions ===<br />
<wikitex>;<br />
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=<br />
\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\mbox{\ for\ }f\in F,$$<br />
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.<br />
<br />
In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.<br />
<br />
The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. <br />
If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. <br />
</wikitex><br />
<br />
=== Submersions ===<br />
<wikitex>;<br />
Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. <br />
[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]<br />
An example of a submersion, which is not a fiber bundle, is given by<br />
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$<br />
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$<br />
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.<br />
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$<br />
for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the M&ouml;bius band). Their leaf spaces are not Hausdorff.<br />
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]<br />
</wikitex><br />
<br />
=== Reeb foliations ===<br />
<wikitex>;<br />
Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by<br />
$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$<br />
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ <br />
for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.<br />
<br />
</wikitex><br />
<br />
=== Taut foliations ===<br />
<wikitex>;<br />
A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. <br />
<br />
{{beginthm|Theorem|(Rummler, Sullivan)}}<br />
The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$:<br />
<br />
a) $\mathcal{F}$ is taut;<br />
<br />
b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$;<br />
<br />
c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}} <br />
<br />
<br />
</wikitex><br />
<br />
=== Constructing new foliations from old ones ===<br />
<br />
==== Pullbacks ====<br />
<wikitex>;<br />
{{beginthm|Theorem |}} If $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ and $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$, then $N$ is foliated by connected components of $f^{-1}\left(L\right)$ as $L$ ranges over the leaves of $\mathcal{F}$. {{endthm}}<br />
{{cite|Candel&Conlon2000}}, Theorem 3.2.2<br />
</wikitex><br />
<br />
==== Glueing ====<br />
<wikitex>;<br />
Let $\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$ be $n$-manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$.<br />
If either both foliations are tangent or both foliations are transverse to the boundaries of $M_1$ and $M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$. This is called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$.<br />
</wikitex><br />
<br />
==== Turbulization ====<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$. <br />
<br />
Define a foliation ${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by<br />
$$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$<br />
where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$<br />
The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhood of the boundary of $N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$ of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<br />
=== Holonomy ===<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation and $L$ a leaf. For a path $\gamma:\left[0,1\right]\rightarrow L$ contained in the intersection of the leaf $L$ with<br />
a foliation chart $U$, and two transversals $\tau_0,\tau_1$ to $\gamma$ at the endpoints, the product structure of the foliation chart determines a homeomorphism $$h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$$<br />
If $\gamma$ is covered by foliation charts $U_0,\ldots,U_k$, then one obtains a sequence of homeomorphisms $$h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$$<br />
The composition yields a well-defined map $h$ from the germ of $\tau_0$ at $\gamma\left(0\right)$<br />
to the germ of $\tau_{k+1}$ at $\gamma\left(1\right)$, the so-called holonomy transport. The holonomy transport only depends on the relative homotopy class of $\gamma$. <br />
{{beginthm|Lemma}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation, $L$ a leaf, $x\in L$ and $\tau$ a transversal at $x$. Holonomy transport defines a homomorphism $$H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)$$ to the group of germs of homeomorphisms of $\tau$.<br />
{{endthm}}<br />
<br />
{{beginthm|Corollary|(Reeb)}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension one foliation of a 3-manifold such that some leaf $L$ is homeomorphic to $S^2$. Then $M=S^2\times S^1$ and $\mathcal{F}$ is the product foliation by spheres.<br />
{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.5<br />
<br />
<br />
</wikitex><br />
=== Godbillon-Vey invariant ===<br />
<wikitex>;<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation of a manifold $M$, then its tangential plane field $E$ is defined by a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ and $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. The Godbillon-Vey invariant of $\mathcal{F}$ is defined as<br />
$$gv\left({\mathcal{F}}\right):=\left[\eta\wedge\left(d\eta\right)^q\right]\in H^{2q+1}_{dR}\left(M\right).$$<br />
<br />
The Godbillion-Vey invariant is related to resilience of leaves. A leaf is said to be resilient if it is not properly embedded and its holonomy is not trivial.<br />
<br />
{{beginthm|Theorem|(Duminy)}}<br />
If $\left(M,{\mathcal{F}}\right)$ is a foliation of codimension one and no leaf is resilient, then $gv\left({\mathcal{F}}\right)=0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== Classification ==<br />
<br />
=== Codimension one foliations ===<br />
<br />
==== Existence ====<br />
<wikitex>;<br />
{{beginthm|Theorem}}<br />
A closed smooth manifold $M^n$ has a smooth codimension one foliation if and only if $\chi(M^n)=0$, where $\chi$ denotes the Euler characteristic. <br />
<br />
If $\chi(M^n)=0$, then every $(n-1)$-plane field $\tau^{n-1}$ on $M^n$ is homotopic to the tangent plane field of a smooth codimension one foliation.{{endthm}}<br />
<br />
{{cite|Thurston1976}}<br />
</wikitex><br />
<br />
==== Foliations of surfaces ====<br />
<wikitex>;<br />
If $\left({\mathbb R}^2,{\mathcal{F}}\right)$ is a codimension one foliation of the plane ${\mathbb R}^2$, then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold $L$. This provides a 1-1-correspondence between foliations of ${\mathbb R}^2$ and simply connected 1-manifolds. <br />
<br />
Codimension one foliations on compact surfaces $S$ exist only if $\chi\left(S\right)=0$, that is on the Torus, the Klein bottle, the annulus and the Möbius band.<br />
<br />
A foliation $\left(S,{\mathcal{F}}\right)$ is said to contain a Reeb component resp. a non-orientable Reeb component if the restriction of ${\mathcal{F}}$ to some subsurface $S^\prime\subset S$ is a Reeb foliation resp. a non-orientable Reeb foliation. (This implies that $S^\prime$ is an annulus resp. a Möbius band.)<br />
{{beginthm|Theorem}} <br />
<br />
a) Let $\left(S,{\mathcal{F}}\right)$ be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:<br />
<br />
(1) $\mathcal{F}$ is the suspension of a homeomorphism $f:S^1\rightarrow S^1$ or<br />
<br />
(2) $\mathcal{F}$ contains a Reeb component (orientable or not).<br />
<br />
b) Every foliation of the annulus $S^1\times I$ tangent to the boundary is obtained by glueing together a finite number of Reeb components and a finite number of suspensions<br />
<br />
c) Every foliation of the Möbius band tangent to the boundary is one of the following three possibly glued together with a foliation on $S^1\times I$:<br />
<br />
(1) the non-orientable Reeb component<br />
<br />
(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution<br />
<br />
(3) the suspension of an orientation-reversing homeomorpism $f:I\rightarrow I$.{{endthm}}<br />
{{cite|Hector&Hirsch1981}}, Theorem 4.2.15 and Proposition 4.3.2<br />
</wikitex><br />
<br />
==== Foliations of 3-manifolds ====<br />
<wikitex>;<br />
{{beginthm|Theorem|(Novikov)}} If a 3-manifold $M$ admits a foliation $\mathcal{F}$ without Reeb components, then $\pi_2\left(M\right)=0$, every leaf of $\mathcal{F}$ is incompressible, and every transverse loop is essential in $\pi_1\left(M\right)$. {{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.37<br />
<br />
A taut foliation has no Reeb component. If $M$ is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.<br />
<br />
{{beginthm|Theorem|(Palmeira)}} If $\mathcal{F}$ is a taut foliation of a 3-manifold $M$ not finitely covered by $S^2\times S^1$, then the universal covering $\widetilde{M}$ is homeomorphic to ${\mathbb R}^3$ and the pull-back foliation $\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$ is homeomorphic to a product foliation $\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$, where $\mathcal{G}$ is a foliation of ${\mathbb R}^2$ by lines.{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.38<br />
<br />
In particular, a taut foliation of a 3-manifold $M$ yields an action of $\pi_1M$ on a (possibly non-Hausdorff) simply connected 1-manifold $L$, the space of leaves of ${\mathcal{G}}$.<br />
<br />
{{beginthm|Theorem|(Gabai)}} Let $M$ be a closed, irreducible 3-manifold.<br />
<br />
a) If $H_2\left(M;{\mathbb R}\right)\not =0$, then $M$ admits a taut foliation.<br />
<br />
b) If $S$ is a surface which minimizes the Thurston norm in its homology class $\left[S\right]\in H_2\left(M;{\mathbb R}\right)$, then $M$ admits a taut foliaton for which $S$ is a leaf.{{endthm}}<br />
<br />
=== Codimension two foliations ===<br />
<br />
==== $S^1$-foliations of 3-manifolds ====<br />
<br />
{{beginthm|Theorem|(Epstein)}} Every foliation of a compact 3-manifold by circles is a Seifert fibration.{{endthm}}<br />
<br />
{{beginthm|Example}}<br />
a) For every rational number $\frac{p}{q}\not=0$ there exists a foliaton of $S^3=\left\{\left(z,w\right)\in{\mathbb C}^2: \mid z\mid^2+\mid w\mid^2=1\right\}$ by circles such that restriction to the standard embedded torus $\left\{\left(z,w\right)\in S^3: \mid z\mid=\mid w\mid=1\right\}$ is the suspension foliation of $R_{\frac{p}{q}2\pi}$. <br />
b) The complement of a knot $K\subset S^3$ admits a foliation by circles if and only if $K$ is a torus knot.{{endthm}}<br />
<br />
{{beginthm|Theorem|(Vogt)}} If a 3-manifold $M$ admits a foliation by circles, then any 3-manifold obtained by removing finitely many points from $M$ admits a (not necessarily smooth) foliation by circles.{{endthm}}<br />
<br />
{{beginthm|Corollary}} ${\mathbb R}^3$ admits a foliation by circles.{{endthm}}<br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
{{Stub}}</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationsFoliations2012-09-20T11:23:31Z<p>Kuessner: /* Bundles */</p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
=== Bundles ===<br />
<br />
<wikitex>;<br />
<br />
The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)<br />
<br />
A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)<br />
<br />
Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a diffeomorphism $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:<br />
$$\begin{xy}<br />
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\<br />
U\ar[r]^{id}&U}<br />
\end{xy}$$<br />
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.<br />
</wikitex><br />
<br />
=== Suspensions ===<br />
<wikitex>;<br />
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=<br />
\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\mbox{\ for\ }f\in F,$$<br />
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.<br />
<br />
In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.<br />
<br />
The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. <br />
If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. <br />
</wikitex><br />
<br />
=== Submersions ===<br />
<wikitex>;<br />
Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. <br />
[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]<br />
An example of a submersion, which is not a fiber bundle, is given by<br />
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$<br />
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$<br />
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.<br />
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$<br />
for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the M&ouml;bius band). Their leaf spaces are not Hausdorff.<br />
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]<br />
</wikitex><br />
<br />
=== Reeb foliations ===<br />
<wikitex>;<br />
Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by<br />
$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$<br />
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ <br />
for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.<br />
<br />
</wikitex><br />
<br />
=== Taut foliations ===<br />
<wikitex>;<br />
A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. <br />
<br />
{{beginthm|Theorem|(Rummler, Sullivan)}}<br />
The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$:<br />
<br />
a) $\mathcal{F}$ is taut;<br />
<br />
b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$;<br />
<br />
c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}} <br />
<br />
<br />
</wikitex><br />
<br />
=== Constructing new foliations from old ones ===<br />
<br />
==== Pullbacks ====<br />
<wikitex>;<br />
{{beginthm|Theorem |}} If $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ and $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$, then $N$ is foliated by connected components of $f^{-1}\left(L\right)$ as $L$ ranges over the leaves of $\mathcal{F}$. {{endthm}}<br />
{{cite|Candel&Conlon2000}}, Theorem 3.2.2<br />
</wikitex><br />
<br />
==== Glueing ====<br />
<wikitex>;<br />
Let $\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$ be $n$-manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$.<br />
If either both foliations are tangent or both foliations are transverse to the boundaries of $M_1$ and $M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$. This is called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$.<br />
</wikitex><br />
<br />
==== Turbulization ====<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$. <br />
<br />
Define a foliation ${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by<br />
$$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$<br />
where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$<br />
The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhood of the boundary of $N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$ of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<br />
=== Holonomy ===<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation and $L$ a leaf. For a path $\gamma:\left[0,1\right]\rightarrow L$ contained in the intersection of the leaf $L$ with<br />
a foliation chart $U$, and two transversals $\tau_0,\tau_1$ to $\gamma$ at the endpoints, the product structure of the foliation chart determines a homeomorphism $$h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$$<br />
If $\gamma$ is covered by foliation charts $U_0,\ldots,U_k$, then one obtains a sequence of homeomorphisms $$h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$$<br />
The composition yields a well-defined map $h$ from the germ of $\tau_0$ at $\gamma\left(0\right)$<br />
to the germ of $\tau_{k+1}$ at $\gamma\left(1\right)$, the so-called holonomy transport. The holonomy transport only depends on the relative homotopy class of $\gamma$. <br />
{{beginthm|Lemma}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation, $L$ a leaf, $x\in L$ and $\tau$ a transversal at $x$. Holonomy transport defines a homomorphism $$H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)$$ to the group of germs of homeomorphisms of $\tau$.<br />
{{endthm}}<br />
<br />
{{beginthm|Corollary|(Reeb)}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension one foliation of a 3-manifold such that some leaf $L$ is homeomorphic to $S^2$. Then $M=S^2\times S^1$ and $\mathcal{F}$ is the product foliation by spheres.<br />
{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.5<br />
<br />
<br />
</wikitex><br />
=== Godbillon-Vey invariant ===<br />
<wikitex>;<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation of a manifold $M$, then its tangential plane field $E$ is defined by a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ and $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. The Godbillon-Vey invariant of $\mathcal{F}$ is defined as<br />
$$gv\left({\mathcal{F}}\right):=\left[\eta\wedge\left(d\eta\right)^q\right]\in H^{2q+1}_{dR}\left(M\right).$$<br />
<br />
The Godbillion-Vey invariant is related to resilience of leaves. A leaf is said to be resilient if it is not properly embedded and its holonomy is not trivial.<br />
<br />
{{beginthm|Theorem|(Duminy)}}<br />
If $\left(M,{\mathcal{F}}\right)$ is a foliation of codimension one and no leaf is resilient, then $gv\left({\mathcal{F}}\right)=0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== Classification ==<br />
<br />
=== Codimension one foliations ===<br />
<br />
==== Existence ====<br />
<wikitex>;<br />
{{beginthm|Theorem}}<br />
A closed smooth manifold $M^n$ has a smooth codimension one foliation if and only if $\chi(M^n)=0$, where $\chi$ denotes the Euler characteristic. <br />
<br />
If $\chi(M^n)=0$, then every $(n-1)$-plane field $\tau^{n-1}$ on $M^n$ is homotopic to the tangent plane field of a smooth codimension one foliation.{{endthm}}<br />
<br />
{{cite|Thurston1976}}<br />
</wikitex><br />
<br />
==== Foliations of surfaces ====<br />
<wikitex>;<br />
If $\left({\mathbb R}^2,{\mathcal{F}}\right)$ is a codimension one foliation of the plane ${\mathbb R}^2$, then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold $L$. This provides a 1-1-correspondence between foliations of ${\mathbb R}^2$ and simply connected 1-manifolds. <br />
<br />
Codimension one foliations on compact surfaces $S$ exist only if $\chi\left(S\right)=0$, that is on the Torus, the Klein bottle, the annulus and the Möbius band.<br />
<br />
A foliation $\left(S,{\mathcal{F}}\right)$ is said to contain a Reeb component resp. a non-orientable Reeb component if the restriction of ${\mathcal{F}}$ to some subsurface $S^\prime\subset S$ is a Reeb foliation resp. a non-orientable Reeb foliation. (This implies that $S^\prime$ is an annulus resp. a Möbius band.)<br />
{{beginthm|Theorem}} <br />
<br />
a) Let $\left(S,{\mathcal{F}}\right)$ be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:<br />
<br />
(1) $\mathcal{F}$ is the suspension of a homeomorphism $f:S^1\rightarrow S^1$ or<br />
<br />
(2) $\mathcal{F}$ contains a Reeb component (orientable or not).<br />
<br />
b) Every foliation of the annulus $S^1\times I$ tangent to the boundary is obtained by glueing together a finite number of Reeb components and a finite number of suspensions<br />
<br />
c) Every foliation of the Möbius band tangent to the boundary is one of the following three possibly glued together with a foliation on $S^1\times I$:<br />
<br />
(1) the non-orientable Reeb component<br />
<br />
(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution<br />
<br />
(3) the suspension of an orientation-reversing homeomorpism $f:I\rightarrow I$.{{endthm}}<br />
{{cite|Hector&Hirsch1981}}, Theorem 4.2.15 and Proposition 4.3.2<br />
</wikitex><br />
<br />
==== Foliations of 3-manifolds ====<br />
<wikitex>;<br />
{{beginthm|Theorem|(Novikov)}} If a 3-manifold $M$ admits a foliation $\mathcal{F}$ without Reeb components, then $\pi_2\left(M\right)=0$, every leaf of $\mathcal{F}$ is incompressible, and every transverse loop is essential in $\pi_1\left(M\right)$. {{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.37<br />
<br />
A taut foliation has no Reeb component. If $M$ is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.<br />
<br />
{{beginthm|Theorem|(Palmeira)}} If $\mathcal{F}$ is a taut foliation of a 3-manifold $M$ not finitely covered by $S^2\times S^1$, then the universal covering $\widetilde{M}$ is homeomorphic to ${\mathbb R}^3$ and the pull-back foliation $\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$ is homeomorphic to a product foliation $\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$, where $\mathcal{G}$ is a foliation of ${\mathbb R}^2$ by lines.{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.38<br />
<br />
In particular, a taut foliation of a 3-manifold $M$ yields an action of $\pi_1M$ on a (possibly non-Hausdorff) simply connected 1-manifold $L$, the space of leaves of ${\mathcal{G}}$.<br />
<br />
{{beginthm|Theorem|(Gabai)}} Let $M$ be a closed, irreducible 3-manifold.<br />
<br />
a) If $H_2\left(M;{\mathbb R}\right)\not =0$, then $M$ admits a taut foliation.<br />
<br />
b) If $S$ is a surface which minimizes the Thurston norm in its homology class $\left[S\right]\in H_2\left(M;{\mathbb R}\right)$, then $M$ admits a taut foliaton for which $S$ is a leaf.{{endthm}}<br />
<br />
=== Codimension two foliations ===<br />
<br />
==== $S^1$-foliations of 3-manifolds ====<br />
<br />
{{beginthm|Theorem|(Epstein)}} Every foliation of a compact 3-manifold by circles is a Seifert fibration.{{endthm}}<br />
<br />
{{beginthm|Example}}<br />
a) For every rational number $\frac{p}{q}\not=0$ there exists a foliaton of $S^3=\left\{\left(z,w\right)\in{\mathbb C}^2: \mid z\mid^2+\mid w\mid^2=1\right\}$ by circles such that restriction to the standard embedded torus $\left\{\left(z,w\right)\in S^3: \mid z\mid=\mid w\mid=1\right\}$ is the suspension foliation of $R_{\frac{p}{q}2\pi}$. <br />
b) The complement of a knot $K\subset S^3$ admits a foliation by circles if and only if $K$ is a torus knot.{{endthm}}<br />
<br />
{{beginthm|Theorem|(Vogt)}} If a 3-manifold $M$ admits a foliation by circles, then any 3-manifold obtained by removing finitely many points from $M$ admits a (not necessarily smooth) foliation by circles.{{endthm}}<br />
<br />
{{beginthm|Corollary}} ${\mathbb R}^3$ admits a foliation by circles.{{endthm}}<br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
{{Stub}}</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Exotic_spheresExotic spheres2012-03-23T16:17:33Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
By a homotopy sphere $\Sigma^n$ we mean a closed smooth oriented n-manifold homotopy equivalent to $S^n$. The manifold $\Sigma^n$ is called an exotic sphere if it is not diffeomorphic to $S^n$. By the Generalised Poincaré Conjecture proven by Smale, every homotopy sphere in dimension $n \geq 5$ is homeomorphic to $S^n$: this statement holds in dimension 2 by the classification of [[Surfaces|surfaces]] and was famously proven in dimension 4 in {{cite|Freedman1982}} and in dimension 3 by Perelman. We define <br />
$$\Theta_{n} := \{[\Sigma^n] | \Sigma^n \simeq S^n \}$$<br />
to be the set of oriented diffeomorphism classes of homotopy spheres. [[Wikipedia:Connected_sum|Connected sum]] makes $\Theta_n$ into an abelian group with inverse given by reversing orientation. An important subgroup of $\Theta_n$ is $bP_{n+1}$ which consists of those homotopy spheres which bound parallelisable manifolds.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
The first exotic spheres discovered were certain 3-sphere bundles over the 4-sphere, {{cite|Milnor1956}}. Following this discovery there was a rapid development of techniques which construct exotic spheres. We review four such constructions: plumbing, Brieskorn varieties, sphere-bundles and twisting. <br />
</wikitex><br />
<br />
=== Plumbing ===<br />
<wikitex>;<br />
As special case of the following construction goes back at least to {{cite|Milnor1959}}.<br />
<br />
Let $i \in \{1, \dots, n\}$, let $(p_i, q_i)$ be pairs of positive integers such that $p_i + q_i + 2 = n$ and let $\alpha_i \in \pi_{p_i}(SO(q_i+1))$ be the clutching functions of $D^{q_i+1}$-bundles over $S^{p_i + 1}$<br />
$$ D^{q_i+1} \to D(\alpha_i) \to S^{p_i+1}.$$<br />
Let $G$ be a graph with vertices $\{v_1, \dots, v_n\}$ such that the edge set between $v_i$ and $v_j$, is non-empty only if $p_i = q_j$. We form the manifold $W = W(G;\{\alpha_i\})$ from the disjoint union of the $D(\alpha_i)$ by identifying $D^{p_i+1} \times D^{q_i+1}$ and $D^{q_j+1} \times D^{p_j+1}$ for each edge in $G$. If $G$ is simply connected then <br />
$$\Sigma(G, \{\alpha_i \}) : = \partial W$$<br />
is often a homotopy sphere. We establish some notation for graphs, bundles and define <br />
* let $T$ denote the graph with two vertices and one edge connecting them and define $\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$,<br />
* let $E_8$ denote the $E_8$-graph,<br />
* let $\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$-sphere,<br />
* let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$, $k > 4s$, denote a generator,<br />
* let $\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$, denote a generator: <br />
* let $S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$ be the suspension homomorphism,<br />
**$S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$ for $k = 1, 2$ and $S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$ for $k > 2$,<br />
* let $\eta_n : S^{n+1} \to S^n$ be essential.<br />
<br />
Then we have the following exotic spheres.<br />
<br />
* $\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$, the Milnor sphere, generates $bP_{4k}$, $k>1$.<br />
* $\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$, the Kervaire sphere, generates $bP_{4k+2}$.<br />
* $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$. <br />
**For general $k$, $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic.<br />
* $\Sigma^8(\gamma_3^5, \eta_3\tau_4)$, generates $\Theta_8 = \Zz_2$. <br />
* $\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$, generates $\Theta_{16} = \Zz_2$. <br />
</wikitex><br />
<br />
=== Brieskorn varieties ===<br />
<wikitex>;<br />
Let $z = (z_0, \dots , z_n)$ be a point in $\Cc^{n+1}$ and let $a = (a_0, \dots, a_n)$ be a string of n+1 positive integers. Given the complex variety $V(a) : = \{z \, | \, z_0^{a_0} + \dots + z_n^{a_n} =0 \}$ and the $\epsilon$-sphere $S^{2n+1}_\epsilon : = \{ z \, | \, \Sigma_{i=0}^n z_i\bar z_i = \epsilon \}$ for small $\epsilon$, following {{cite|Milnor1968}}<br />
we define the closed smooth oriented $(n-2)$-connected $(2n-1)$-manifold<br />
$$ W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.$$<br />
The manifolds $W^{2n-1}(a)$ are often called Brieskorn varieties. By construction, every $W^{2n-1}(a)$ lies in $S^{2n+1}$ and so bounds a parallelisable manifold. In {{cite|Brieskorn1966}} and {{cite|Brieskorn1966a}} (see also {{cite|Hirzebruch&Mayer1968}}), it is shown in particular that all homotopy spheres in $bP_{4k}$ and $bP_{4k-2}$ can be realised as $W(a)$ for some $a$. Let $2, \dots, 2$ be a string of 2k-1 2's in a row with $k \geq 2$, then there are diffeomorphisms<br />
$$ W^{4k-1}(3, 6r-1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},$$<br />
$$ W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.$$<br />
</wikitex><br />
<br />
=== Sphere bundles ===<br />
<wikitex>;<br />
The first known examples of exotic spheres were discovered by Milnor in {{cite|Milnor1956}}. They are the total spaces of certain 3-[[Wikipedia:Sphere_bundle#Sphere_bundles|sphere bundles]] over the 4-sphere as we now explain: the group $\pi_3(SO(4)) \cong \Zz \oplus \Zz$ parametrises linear $3$-sphere bundles over $S^4$ where a pair $(m, n)$ gives rise to a bundle with Euler number $n$ and first Pontrjagin class $2(n+2m)$: here we orient $S^4$ and so identify $H^4(S^4; \Zz) = \Zz$. If we set $n = 1$ then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold $\Sigma^7_{m, 1}$, the total space of the bundle $(m, 1)$, is a homotopy sphere. Milnor first used a $\Zz_7$-invariant, called the $\lambda$-invariant, to show, e.g. that $\Sigma^7_{1, 2}$ is not diffeomorphic to $S^7$. A little later Kervaire and Milnor {{cite|Kervaire&Milnor1963}} proved that $\Theta_7 \cong \Zz_{28}$ and Eells and Kuiper {{cite|Eells&Kuiper1962}} defined a refinement of the $\lambda$-invariant, now called the Eells-Kuiper $\mu$-invariant, which in particular gives<br />
$$ \Sigma^7_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \in bP_8 \cong \Theta_7.$$<br />
<br />
Shimada {{cite|Shimada1957}} used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres. In this case $\pi_7(SO(8)) \cong \Zz \oplus \Zz$ and the bundle $(m, n)$ has Euler number $n$ and second Pontrjagin class $6(n+2m)$. Moreover $\Theta_{15} \cong \Zz_{8,128} \oplus \Zz_2$ where the $\Zz_{8,128}$-summand is $bP_{16}$ as explained below. Results of {{cite|Wall1962a}} and {{cite|Eells&Kuiper1962}} combine to show that<br />
$$ \Sigma^{15}_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.$$<br />
<br />
*By Adams' solution of the [[Wikipedia:Hopf_invariant| Hopf-invariant]] 1 problem, {{cite|Adams1958}} and {{cite|Adams1960}}, the dimensions n = 3, 7 and 15 are the only dimensions in which a topological n-sphere can be fibre over an m-sphere for 0 < m < n. <br />
</wikitex><br />
<br />
=== Twisting ===<br />
<wikitex>;<br />
By {{cite|Cerf1970}} and {{cite|Smale1962a}} there is an isomorphism $\Theta_{n+1} \cong \Gamma_{n+1}$ for $n \geq 5$ where $\Gamma_{n+1} = \pi_0(\Diff_+(S^n))$ is the group of isotopy classes of orientation preserving diffeomorphisms of $S^n$. The map is given by<br />
$$ \Gamma_{n+1} \to \Theta_{n+1}, ~~~~~[f] \longmapsto \Sigma_{f} := D^{n+1} \cup_f (-D^{n+1}).$$<br />
Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of $S^n$ which are not isotopic to the identity. We give such a construction which probably goes back to Milnor: so far the earliest reference found is the problem list of the 1963 Seattle topology conference {{cite|Lashof1965}}.<br />
<br />
Represent $\alpha \in \pi_p(SO(q))$ and $\beta \in \pi_q(SO(p))$ by smooth compactly supported functions $\alpha : \Rr^p \to SO(q)$ and $\beta : \Rr^q \to SO(p)$ and define the following self-diffeomorphisms of $\Rr^p \times \Rr^q$<br />
$$ F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),$$<br />
$$ F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),$$<br />
$$s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.$$<br />
If follows that $s(\alpha, \beta)$ is compactly supported and so extends uniquely to a diffeomorphism of $S^{p+q}$. In this way we obtain a bilinear pairing<br />
$$ \sigma : \pi_q(SO(q)) \otimes \pi_q(SO(p)) \longrightarrow \Theta_{p+q+1}, ~~~~(\alpha, \beta) \longmapsto \Sigma_{s(\alpha, \beta)}$$<br />
such that<br />
$$ \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta).$$<br />
In particular for $k=1, 2$ we see that $\sigma(\gamma_{4k-1}', \gamma_{4k-1}') \cong -\Sigma_M$ generates $bP_{4k}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
Finding invariants of exotic sphere $\Sigma$ which distinguish it from the standard sphere is rather a subtle undertaking. Moreover such invariants are often defined via a manifold $W$ with $\partial W \cong \Sigma$. In this case finding an intrinsic definition and or computation of the relevant invariant can also be subtle.<br />
<br />
We begin by listing some invariants which are equal for all exotic spheres.<br />
{{beginthm|Proposition}} <br />
Let $\Sigma$ be a closed smooth manifold homeomorphic to the n-sphere. Then<br />
# there is an isomorphism of tangent bundles $T\Sigma \cong TS^n$,<br />
# the signature of $\Sigma$ vanishes,<br />
# the Kervaire invariant of $(\Sigma, F)$ is zero for every framing of $\Sigma$.<br />
(To make sense of the first statement remember that the topological space underlying every exotic sphere is homeomorphic to $S^n$.)<br />
{{endthm}}<br />
{{beginrem|Remark}}<br />
The analogue of the first statement for the stable tangent bundle was proven in \cite{Kervaire&Milnor1963|Theorem 3.1}. A proof of the unstable statement is given in \cite{Ray&Pedersen1980|Lemma 1.1}. The next two statements are obvious since both the signature and Kervaire invariant are defined to be zero if $n = 2k+1$ and via a symmetric or quadratic form on $H_k(\Sigma; \Zz) = 0 $ if $n = 2k$.<br />
{{endrem}} <br />
</wikitex><br />
=== Bordism classes ===<br />
<wikitex>;<br />
As every homotopy sphere is stably parallelisable, homotopy spheres admit [[B-Bordism|$B$-structures]] for any $B$. If $B$ is such that $[S^n, F] \mapsto 0 \in \Omega_n^B$ for any stable framing $F$ of $S^n$, then we obtain a well-defined homomorphism<br />
$$ \eta^B : \Theta_n \longrightarrow \Omega_n^B, ~~~\Sigma \longmapsto [\Sigma, F].$$ <br />
* If $B = BO\langle k \rangle$ for $[n/2] + 1 < k < n+2$ then $\Omega_n^B$ is isomorphic to almost framed bordism and the homomorphism $\eta^B$ is the same thing as the $\eta: \Theta_n \to \pi_n(G/O)$ in Theorem \ref{thm-ses}.<br />
* Perhaps surprisingly $\eta_n^{\Spin} \neq 0$ for all $n = 8k+1, 8k+2$, as explained in the next subsection.<br />
* In general determining $\eta^B$ is a hard an interesting problem. <br />
* $B$-coboundaries for elements of $Ker(\eta^B_n)$ are often used to define invariants of $B$-null bordant homotopy spheres.<br />
</wikitex><br />
<br />
=== The α-invariant ===<br />
<wikitex>;<br />
In dimensions $n > 1$, every exotic sphere $\Sigma$ has a unique Spin structure and from above we have the homomorphism $\eta_n^{\Spin} : \Theta_n \to \Omega_n^{\Spin}$. Recall the $\alpha$-invariant homomorphism $\alpha : \Omega_*^{\Spin} \to KO^{-*}$ and that there are isomorphisms $KO^{-8k-1} \cong KO^{-8k-2} \cong \Zz/2$ for all $k \geq 1$.<br />
{{beginthm|Theorem|\cite{Anderson&Brown&Peterson1967}}}<br />
We have $\eta_n^{\Spin}(\Sigma) = 0$ if and only if $\alpha \circ \eta_n^{\Spin}(\Sigma) = 0$ and $\eta_n^{\Spin} \neq 0$ if and only if $n = 8k+1$ or $8k+2$. <br />
{{endthm}}<br />
{{beginrem|Remark}}<br />
Exotic spheres $\Sigma$ with $\alpha(\Sigma) \neq 0$ are often called Hitchin spheres, after \cite{Hitchin1974}: see the discussion of curvature [[#Curvature on exotic spheres|below]].<br />
{{endrem}}<br />
</wikitex><br />
<br />
=== The Eells-Kuiper invariant === <br />
<wikitex>;<br />
</wikitex><br />
=== The s-invariant ===<br />
<wikitex>;<br />
</wikitex><br />
<br />
== Classification ==<br />
<wikitex>;<br />
For $n =1, 2$ and $3$, $\Theta_n = \{ S^n \}$. For $n = 4$, $\Theta_4$ is unknown. We therefore concentrate on higher dimensions.<br />
<br />
For $n \geq 5$, the group of exotic n-spheres $\Theta_n$ fits into the following long exact sequence, first discovered in {{cite|Kervaire&Milnor1963}} (more details can also be found in {{cite|Levine1983}} and {{cite|Lück2001}}):<br />
$$ \dots \stackrel{\eta_{n+1}}{\longrightarrow} \pi_{n+1}(G/O) \stackrel{\sigma_{n+1}}{\longrightarrow} L_{n+1}(e) \stackrel{\omega_{n+1}}{\longrightarrow} \Theta_n \stackrel{\eta_n}{\longrightarrow} \pi_n(G/O) \stackrel{\sigma_n}{\longrightarrow} L_n(e) \to \dots~.$$<br />
Here $L_n(e)$ is the n-th [[Wikipedia:L-theory|L-group]] of the the trivial group: $L_n(e) = \Zz, 0, \Zz/2, 0$ as n = 0, 1, 2 or 3 modulo 4 and the sequence ends at $L_5(e) = 0$. Also $O$ is the stable orthogonal group and $G$ is the stable group of homtopy self-equivalences of the sphere. There is a fibration $O \to G \to G/O$ and the groups $\pi_n(G/O)$ fit into the homtopy long exact sequence<br />
$$ \dots \to \pi_n(O) \to \pi_n(G) \to \pi_n(G/O) \to \pi_{n-1}(O) \to \pi_{n-1}(G) \to \dots $$<br />
of this fibration. The homomorphism $J_n: \pi_n(O) \to \pi_n^S \cong \pi_n(G)$ is the [[Wikipedia:J-homomorphism|stable J-homomorphism]]. In particular, by {{cite|Serre1951}} the groups $\pi_i(G)$ are finite and by {{cite|Bott1959}}, {{cite|Adams1966}} and {{cite|Quillen1971}} the domain, image and kernel of $J_n$ have been completely determined. An important result in {{cite|Kervaire&Milnor1963}} is that the homomorphism $\sigma_{4k}$ is nonzero. The above sequence then gives<br />
<br />
{{beginthm|Theorem|{{cite|Kervaire&Milnor1963}}}}\label{thm-ses}<br />
For $n \geq 5$, the group $\Theta_n$ is finite. Moreover there is an exact sequence<br />
$$ 0 \longrightarrow bP_{n+1} \longrightarrow \Theta_{n} \longrightarrow Coker(J_n) \stackrel{K_n}{\longrightarrow} C_n \longrightarrow 0 $$<br />
where $bP_{n+1} := {Im}(\omega_{n+1})$, the group of homotopy spheres bounding paralellisable manifolds, is a finite cyclic group which vanishes if $n$ is even. Moreover $C_n = 0$ unless $n = 4k+2$ when it is $0$ or $\Zz/2$.<br />
{{endthm}}<br />
<br />
The groups $Coker(J_n)$ are known for $n$ up to approximately 62. In general their determination is a very hard problem. Modulo this problem we see two remaining problems in the determination of $\Theta_n$: an extension problem and the comptutation of the order of the groups $bP_{n+1}$ and $C_n$. We discuss these in turn.<br />
<br />
{{beginthm|Theorem|{{cite|Brumfiel1968}}, {{cite|Brumfiel1969}}, {{cite|Brumfiel1970}}}}<br />
If $n \neq 2^{j} - 3$ the Kervaire-Milnor extension splits: <br />
$$\Theta_n \cong bP_{n+1} \oplus Ker(K_n).$$<br />
{{endthm}}<br />
<br />
The map $K_{4k+2} : Coker(J_{4k+2}) \to \Zz/2$ is the Kervaire invariant and by definition $C_{4k+2} = Im(K_{4k+2})$. By the long exact sequence above we have<br />
<br />
{{beginthm|Theorem|{{cite|Kervaire&Milnor1963|Section 8}}}}<br />
The group $bP_{4k+2}$ is either $\Zz/2$ or $0$. Moreover the following are equivalent:<br />
* $bP_{4k+2} = 0$,<br />
* the Kervaire sphere $\Sigma^{4k+1}_K$ is diffeomorphic to the standard sphere,<br />
* there is a framed manifold with Kervaire invariant 1: $C_{4k+2} \cong \Zz/2$.<br />
Conversely the following are equivalent:<br />
* $bP_{4k+2} = \Zz/2$,<br />
* the Kervaire sphere $\Sigma^{4k+1}_K$ is not diffeomorphic to the standard sphere,<br />
* there is no framed manifold with Kervaire invariant 1: $C_{4k+2} \cong 0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== The orders of bP<sub>4k</sub> and bP<sub>4k+2</sub> ===<br />
<wikitex>;<br />
The group $bP_{4k}$ is a cyclic group whose order can be determined using the Hirzebruch's signature theorem if one knows the order of $Im(J_{4k-1}) \subset \pi_{4k-1}(G)$. Adams determined the latter group up to a factor of two which was settled by Quillen with a positive solution to the Adams conjecture. <br />
<br />
{{beginthm|Theorem}}<br />
Let $a_k = (3-(-1)^k)/2$, let $B_k$ be the k-th Bernoulli number (topologist indexing) and for $x \in \Qq$ let $Num(x)$ denote the numerator of $x$ expressed in lowest form. Then for $k \geq 2$, the order of $bP_{4k}$ is<br />
$$ t_k = a_k \cdot 2^{2k-2}\cdot (2^{2k-1}-1) \cdot Num(B_k/4k).$$<br />
{{endthm}}<br />
<br />
{{beginrem|Remark}}<br />
Note that $Num(B_k/4k)$ is odd so the 2-primary order of $bP_{4k}$ is $a_k \cdot 2^{2k-2}$ while the odd part is $(2^{2k-1}-1) \cdot Num(B_k/4k)$. Modulo the Adams conjecture the proof appeared in {{cite|Kervaire&Milnor1963|Section 7}}. Detailed treatments can also be found in {{cite|Levine1983|Section 3}} and {{cite|Lück2001|Chapter 6}}.<br />
{{endrem}}<br />
<br />
The next theorem describes the situation for $bP_{4k+2}$ which is now almost completely understood as well. References for the theorem are given in the remark which follows it.<br />
<br />
{{beginthm|Theorem}}<br />
The group $bP_{4k+2}$ is given as follows:<br />
* $bP_{6} = bP_{14} = bP_{30} = bP_{62} = 0$,<br />
* $bP_{126} = 0$ or $\Zz/2$,<br />
* $bP_{4k+2} = \Zz/2$ else.<br />
{{endthm}}<br />
<br />
{{beginrem|Remark}}<br />
The following is a chronological list of determinations of $bP_{4k+2}$:<br />
* $bP_{10} = \Zz/2$, {{cite|Kervaire1960a}}.<br />
* $bP_{6} = bP_{14} = 0$ {{cite|Kervaire&Milnor1963}}.<br />
* $bP_{8k+2} = \Zz/2$, {{cite|Anderson&Brown&Peterson1966a}}.<br />
* $bP_{30} = 0$, {{cite|Mahowald&Tangora1967}}.<br />
* $bP_{4k+2} = \Zz/2$ unless $4k+2 = 2^j - 2$ {{cite|Browder1969}}.<br />
* $bP_{62} = 0$, {{cite|Barratt&Jones&Mahowald1984}}.<br />
* $bP_{2^j - 2} = \Zz/2$ for $j \geq 8$, {{cite|Hill&Hopkins&Ravenel2009}}.<br />
{{endrem}}<br />
</wikitex><br />
<br />
== Further discussion ==<br />
=== Curvature on exotic spheres ===<br />
<wikitex>;<br />
Gromoll-Meyer proved that a certain exotic 7-sphere can be realized as a biquotient of the compact Lie group Sp(2) and thus by the O'Neill formula has a Riemannian metric of nonnegative sectional curvature. It is not known whether there exist exotic spheres with Riemannian metrics of positive sectional curvature. For a recent review of which exotic spheres admit metrics of various sorts of positive curvature see \cite{Joachim&Wraith2008}.<br />
</wikitex><br />
<br />
=== The Kervaire-Milnor braid ===<br />
<wikitex>;<br />
$$<br />
\def\curv{1.5pc}% Adjust the curvature of the curved arrows here<br />
\xymatrix@!R@!C@!0@R=2.5pc@C=4pc{% Adjust the spacing here<br />
\pi_n(\Top/O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(G) \\<br />
& \pi_n(G/O) \ar[dr] \ar[ur] && \pi_{n-1}(\Top) \ar[dr] \ar[ur] \\<br />
\pi_n(G) \ar[ur] \ar@/d\curv/[rr] && \pi_n(G/\Top) \ar[ur] \ar@/d\curv/[rr] && \pi_{n-1}(\Top/O)<br />
}<br />
$$<br />
</wikitex><br />
== PL manifolds admitting no smooth structure ==<br />
<wikitex>;<br />
Let $W^{2n}$ be a [[Exotic spheres#Plumbing|plumbing manifold]] as described above. By a simple version of the [[Alexander trick]], there is a homemorphism $f \colon \partial W \cong S^{2n-1}$ and so we can form the closed topological manifold<br />
$$ \bar W : = W \cup_f D^{2n}.$$<br />
If $\partial W$ is exotic then it turns out that $\bar W$ is a topological manifold which admits no smooth structure! <br />
<br />
\cite{Kervaire1960a} shows that $\bar W^{10}$ is non-smoothable and the arugments there work for all odd $n$ so long as the Kervaire sphere is exotic. <br />
<br />
When $n$ is even the proof is more complicated: one first need's Novikov's theorem that the rational Pontrjagin classes of a topological manifold are homeomorphism invariants \cite{Novikov1965b}. Prior to Novikvo's result, some weaker statements were known. For example, when $n=4$ and $W$ is the total space of a [[Exotic spheres#Sphere bundles|$D^4$-bundle]] over $S^4$ as above and if $\partial W = \Sigma_{m, 1}$ then by \cite{Tamura1961} $\bar W$ is smoothable if and only if $m(m-1)/2 \equiv 0$ mod $4$.</wikitex><ref>Note that Tamura uses a different identification <tex>\pi_3(SO(4)) \cong \Zz \oplus \Zz</tex> from the one used above.</ref><wikitex> Applying Novikov's theorem we know that $\bar W$ is smoothable if and only if $m(m-1)/2 \equiv 0$ mod $56$.<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
== Footnotes ==<br />
<references/><br />
<br />
== External links ==<br />
* The Wikipedia page on [[Wikipedia:Exotic sphere|exotic spheres]]<br />
* The tabulation of the order of the group of exotic spheres in the [http://oeis.org/classic/A001676 On-Line Encyclopedia of Integer Sequences]<br />
* Andrew Ranicki's exotic sphere home page, with many of the original papers: [http://www.maths.ed.ac.uk/~aar/exotic.htm http://www.maths.ed.ac.uk/~aar/exotic.htm] <br />
** Including some [http://www.maths.ed.ac.uk/~aar/papers/km-it.pdf original correspondence between Kervaire and Milnor]<br />
*[http://www.nilesjohnson.net/seven-manifolds.html An animation of exotic 7-spheres]. Slides from a presentation by [http://www.nilesjohnson.net/ Nile Johsnon] at the [http://www.ima.umn.edu/2011-2012/SW1.30-2.1.12/ Second Abel conference] in honor of [[Wikipedia:John Milnor|John Milnor]].<br />
[[Category:Manifolds]]<br />
[[Category:Highly-connected manifolds]]<br />
[[Category:Surgery]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Geometric_3-manifoldsGeometric 3-manifolds2012-03-23T12:06:38Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Let a group $G$ act on a manifold $X$ by homeomorphisms. <br />
<br />
A $\left(G,X\right)$-manifold is a manifold $M$ with a $\left(G,X\right)$-atlas, that is, a collection $\left\{\left(U_i,\phi_i\right):i\in I\right\}$ of homeomorphisms <br />
$$\phi_i:U_i\rightarrow \phi_i\left(U_i\right)\subset X$$ <br />
onto open subsets of $X$ such that all coordinate changes <br />
$$\gamma_{ij}=\phi_i\phi_j^{-1}:\phi_i\left(U_i\cap U_j\right)\rightarrow \phi_j\left(U_i\cap U_j\right)$$<br />
are restrictions of elements of $G$.<br />
<br />
Fix a basepoint $x_0\in M$ and a chart $\left(U_0,\phi_0\right)$ with $x_0\in U_0$. Let $\pi:\widetilde{M}\rightarrow M$ be the universal covering. These data determine the developing map $$D:\widetilde{M}\rightarrow X$$ that agrees with the analytic continuation of $\phi_0\pi$ along each path, in a neighborhood of the path's endpoint.<br />
<br />
If we change the initial data $x_0$ and $\left(U_0,\phi_0\right)$, the developing map $D$ changes by composition with an element of $G$.<br />
<br />
If $\sigma\in\pi_1\left(M,x_0\right)$, analytic continuation along a loop representing $\sigma$ gives a chart $\phi_0^\sigma$ that is comparable to $\phi_0$, since they are both defined at $x_0$. Let $g_\sigma$ be the element of $G$ such that $\phi_0^\sigma=g_\sigma\phi_0$. The map $$H:\pi_1\left(M,x_0\right)\rightarrow G, H\left(\sigma\right)=g_\sigma$$<br />
is a group homomorphism and is called the holonomy of $M$.<br />
<br />
If we change the initial data $x_0$ and $\left(U_0,\phi_0\right)$, the holonomy homomorphisms $H$ changes by conjugation with an element of $G$.<br />
<br />
A $\left(G,X\right)$-manifold is complete if the developing map $D:\widetilde{M}\rightarrow X$ is surjective.<br />
<br />
{{cite|Thurston1997}} Section 3.4<br />
<br />
{{beginthm|Definition}} <br />
A model geometry $\left(G,X\right)$ is a smooth manifold $X$ together with a Lie group of diffeomorphisms of $X$, such that:<br />
<br />
a) $X$ is connected and simply connected;<br />
<br />
b) $G$ acts transitively on $X$, with compact point stabilizers;<br />
<br />
c) $G$ is not contained in any larger group of diffeomorphisms of $X$ with compact point stabilizers;<br />
<br />
d) there exists at least one compact $\left(G,X\right)$-manifold.<br />
{{endthm}}<br />
{{cite|Thurston1997}} Definition 3.8.1<br />
<br />
A 3-manifold is said to be a geometric manifold if it is a $\left(G,X\right)$-manifold for a 3-dimensional model geometry $\left(G,X\right)$.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
{{beginthm|Theorem}}There are eight 3-dimensional model geometries:<br />
<br />
- the round sphere: $X=S^3, G=O(4)$<br />
<br />
- Euclidean space: $X={\mathbb R}^3, G={\mathbb R}^3\rtimes O(3)$<br />
<br />
- hyperbolic space: $X= H^3, G=PSL\left(2,{\mathbb C}\right)\rtimes {\mathbb Z}/2{\mathbb Z}$<br />
<br />
- $X=S^2\times {\mathbb R}, G=O(3)\times \left({\mathbb R}\rtimes {\mathbb Z}/2{\mathbb Z}\right)$<br />
<br />
- $X={\mathbb H}^2\times {\mathbb R}, G=\left(PSL\left(2,{\mathbb R}\right)\rtimes{\mathbb Z}/2{\mathbb Z}\right)\times \left({\mathbb R}\rtimes {\mathbb Z}/2{\mathbb Z}\right)$<br />
<br />
- the universal covering of the unit tangent bundle of the hyperbolic plane: $G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$<br />
<br />
- the Heisenberg group: $G=X=Nil=\left\{\left(\begin{matrix}1&x&z\\0&1&y\\0&0&1\end{matrix}\right):x,y,z\in{\mathbb R}\right\}$<br />
<br />
- the 3-dimensional solvable Lie group $G=X=Sol={\mathbb R}^2\rtimes {\mathbb R}$ with conjugation $t\rightarrow\left(\begin{matrix}e^t&0\\0&e^{-t}\end{matrix}\right)$.{{endthm}}<br />
{{cite|Thurston1997}} Section 3.8<br />
<br />
Outline of Proof:<br />
<br />
Let $G^\prime$ be the connected component of the identity of $G$, and let $G_x^\prime$ be the stabiliser of $x\in X$. <br />
$G^\prime$ acts transitively and $G_x^\prime$ is a closed, connected subgroup of $SO\left(3\right)$.<br />
<br />
Case 1: $G_x^\prime=SO\left(3\right)$. Then $X$ has constant sectional curvature. The Cartan Theorem implies that (up to rescaling) $X$ is isometric to one of $S^3, {\mathbb R}^3, H^3$.<br />
<br />
Case 2: $G_x^\prime \simeq SO\left(2\right)$. Let $V$ be the $G^\prime$-invariant vector field such that, for each $x\in X$, the direction of $V_x$ is the rotation axis of $G_x^\prime$. $V$ descends to a vector field on compact $\left(G,X\right)$-manifolds, therefore the flow of $V$ must preserve volume. In our setting this implies that the flow of $V$ acts by isometries. Hence the flowlines define a 1-dimensional foliation ${\mathcal{F}}$ with embedded leaves. The quotient $X/{\mathcal{F}}$ is a 2-dimensional manifold, which inherits a Riemannian metric such that $G^\prime$ acts transitively by isometries. Thus $Y:=X/{\mathcal{F}}$ has constant curvature and is (up to rescaling) isometric to one of $S^2, {\mathbb R}^2, H^2$. $X$ is a pricipal bundle over $Y$ with fiber ${\mathbb R}$ or $S^1$,<br />
The plane field $\tau$ orthogonal to $\mathcal{F}$ has constant curvature, hence it is either a foliation or a contact structure.<br />
<br />
Case 2a: $\tau$ is a foliation. Thus $X$ is a flat bundle over $Y$. $Y$ is one of $S^2, {\mathbb R}^2, H^2$, hence $\pi_1Y=0$, which implies that $X=Y\times {\mathbb R}$.<br />
<br />
Case 2b: $\tau$ is a contact structure. <br />
For $Y=S^2$ one would obtain for $G$ the group of isometries of $S^3$ that preserve the Hopf fibration. This is not a maximal group with compact stabilizers, thus there is no model geometry in this case.<br />
For $Y={\mathbb R}^2$ one obtains $G=X=Nil$. Namely, $G$ is the subgroup of the group of automorphisms of the standard contact structure $dz-xdy=0$ on ${\mathbb R}^3$ consisting of those automorphisms which are lifts of isometries of the x-y-plane.<br />
For $Y={\mathbb H}^2$ one obtains $G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$.<br />
<br />
Case 3: $G_x^\prime=1$. Then $X=G^\prime/G_x^\prime=G^\prime$ is a Lie group. The only 3-dimensional unimodular Lie group which is not subsumed by one of the previous geometries is $G=Sol$.<br />
<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<br />
A closed 3-manifold is called:<br />
<br />
- irreducible, if every embedded 2-sphere bounds an embedded 3-ball,<br />
<br />
- geometrically atoroidal, if there is no embedded incompressible torus,<br />
<br />
- homotopically atoroidal, if there is no immersed incompressible torus.<br />
<br />
<wikitex>;<br />
{{beginthm|Theorem|(Geometrization)}}<br />
<br />
Let $M$ be a closed, orientable, irreducible, geometrically atoroidal 3-manifold.<br />
<br />
a) If $M$ is homotopically atoroidal, then it admits an $H^3$-geometry.<br />
<br />
b) If $M$ is not homotopically atoroidal, then it admits (at least) one of the seven non-$H^3$-geometries.<br />
{{endthm}}<br />
<br />
{{beginthm|Example|(Geometrization of mapping tori)}} <br />
<br />
Let $\Phi:\Sigma_g\rightarrow \Sigma_g$ be an orientation-preserving homeomorphism of the surface of genus $g$.<br />
<br />
a) If $g=1$, then the mapping torus $M_\Phi$ satisfies the following:<br />
<br />
1. If $\Phi$ is periodic, then $M_\Phi$ admits an ${\mathbb R}^3$ geometry.<br />
<br />
2. If $\Phi$ is reducible, then $M_\Phi$ contains an embedded incompressible torus.<br />
<br />
3. If $\Phi$ is Anosov, then $M_\Phi$ admits a $Sol$ geometry.<br />
<br />
b) If $g\ge 2$, then the mapping torus $M_\Phi$ satisfies the following:<br />
<br />
1. If $\Phi$ is periodic, then $M_\Phi$ admits an $H^2\times{\mathbb R}$-geometry.<br />
<br />
2. If $\Phi$ is reducible, then $M_\Phi$ contains an embedded incompressible torus.<br />
<br />
3. If $\Phi$ is pseudo-Anosov, then $M_\Phi$ admits an $H^3$-geometry.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
As the example suggests, the most abundant case ist that of [[Hyperbolic 3-manifolds]].<br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Exotic_spheresExotic spheres2012-03-23T12:01:02Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
By a homotopy sphere $\Sigma^n$ we mean a closed smooth oriented n-manifold homotopy equivalent to $S^n$. The manifold $\Sigma^n$ is called an exotic sphere if it is not diffeomorphic to $S^n$. By the Generalised Poincaré Conjecture proven by Smale, every homotopy sphere in dimension $n \geq 5$ is homeomorphic to $S^n$: this statement holds in dimension 2 by the classification of [[Surfaces|surfaces]] and was famously proven in dimension 4 in {{cite|Freedman1982}} and in dimension 3 by Perelman. We define <br />
$$\Theta_{n} := \{[\Sigma^n] | \Sigma^n \simeq S^n \}$$<br />
to be the set of oriented diffeomorphism classes of homotopy spheres. [[Wikipedia:Connected_sum|Connected sum]] makes $\Theta_n$ into an abelian group with inverse given by reversing orientation. An important subgroup of $\Theta_n$ is $bP_{n+1}$ which consists of those homotopy spheres which bound parallelisable manifolds.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
The first exotic spheres discovered were certain 3-sphere bundles over the 4-sphere, {{cite|Milnor1956}}. Following this discovery there was a rapid development of techniques which construct exotic spheres. We review four such constructions: plumbing, Brieskorn varieties, sphere-bundles and twisting. <br />
</wikitex><br />
<br />
=== Plumbing ===<br />
<wikitex>;<br />
As special case of the following construction goes back at least to {{cite|Milnor1959}}.<br />
<br />
Let $i \in \{1, \dots, n\}$, let $(p_i, q_i)$ be pairs of positive integers such that $p_i + q_i + 2 = n$ and let $\alpha_i \in \pi_{p_i}(SO(q_i+1))$ be the clutching functions of $D^{q_i+1}$-bundles over $S^{p_i + 1}$<br />
$$ D^{q_i+1} \to D(\alpha_i) \to S^{p_i+1}.$$<br />
Let $G$ be a graph with vertices $\{v_1, \dots, v_n\}$ such that the edge set between $v_i$ and $v_j$, is non-empty only if $p_i = q_j$. We form the manifold $W = W(G;\{\alpha_i\})$ from the disjoint union of the $D(\alpha_i)$ by identifying $D^{p_i+1} \times D^{q_i+1}$ and $D^{q_j+1} \times D^{p_j+1}$ for each edge in $G$. If $G$ is simply connected then <br />
$$\Sigma(G, \{\alpha_i \}) : = \partial W$$<br />
is often a homotopy sphere. We establish some notation for graphs, bundles and define <br />
* let $T$ denote the graph with two vertices and one edge connecting them and define $\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$,<br />
* let $E_8$ denote the $E_8$-graph,<br />
* let $\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$-sphere,<br />
* let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$, $k > 4s$, denote a generator,<br />
* let $\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$, denote a generator: <br />
* let $S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$ be the suspension homomorphism,<br />
**$S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$ for $k = 1, 2$ and $S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$ for $k > 2$,<br />
* let $\eta_n : S^{n+1} \to S^n$ be essential.<br />
<br />
Then we have the following exotic spheres.<br />
<br />
* $\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$, the Milnor sphere, generates $bP_{4k}$, $k>1$.<br />
* $\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$, the Kervaire sphere, generates $bP_{4k+2}$.<br />
* $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$. <br />
**For general $k$, $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic.<br />
* $\Sigma^8(\gamma_3^5, \eta_3\tau_4)$, generates $\Theta_8 = \Zz_2$. <br />
* $\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$, generates $\Theta_{16} = \Zz_2$. <br />
</wikitex><br />
<br />
=== Brieskorn varieties ===<br />
<wikitex>;<br />
Let $z = (z_0, \dots , z_n)$ be a point in $\Cc^{n+1}$ and let $a = (a_0, \dots, a_n)$ be a string of n+1 positive integers. Given the complex variety $V(a) : = \{z \, | \, z_0^{a_0} + \dots + z_n^{a_n} =0 \}$ and the $\epsilon$-sphere $S^{2n+1}_\epsilon : = \{ z \, | \, \Sigma_{i=0}^n z_i\bar z_i = \epsilon \}$ for small $\epsilon$, following {{cite|Milnor1968}}<br />
we define the closed smooth oriented $(n-2)$-connected $(2n-1)$-manifold<br />
$$ W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.$$<br />
The manifolds $W^{2n-1}(a)$ are often called Brieskorn varieties. By construction, every $W^{2n-1}(a)$ lies in $S^{2n+1}$ and so bounds a parallelisable manifold. In {{cite|Brieskorn1966}} and {{cite|Brieskorn1966a}} (see also {{cite|Hirzebruch&Mayer1968}}), it is shown in particular that all homotopy spheres in $bP_{4k}$ and $bP_{4k-2}$ can be realised as $W(a)$ for some $a$. Let $2, \dots, 2$ be a string of 2k-1 2's in a row with $k \geq 2$, then there are diffeomorphisms<br />
$$ W^{4k-1}(3, 6r-1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},$$<br />
$$ W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.$$<br />
</wikitex><br />
<br />
=== Sphere bundles ===<br />
<wikitex>;<br />
The first known examples of exotic spheres were discovered by Milnor in {{cite|Milnor1956}}. They are the total spaces of certain 3-[[Wikipedia:Sphere_bundle#Sphere_bundles|sphere bundles]] over the 4-sphere as we now explain: the group $\pi_3(SO(4)) \cong \Zz \oplus \Zz$ parametrises linear $3$-sphere bundles over $S^4$ where a pair $(m, n)$ gives rise to a bundle with Euler number $n$ and first Pontrjagin class $2(n+2m)$: here we orient $S^4$ and so identify $H^4(S^4; \Zz) = \Zz$. If we set $n = 1$ then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold $\Sigma^7_{m, 1}$, the total space of the bundle $(m, 1)$, is a homotopy sphere. Milnor first used a $\Zz_7$-invariant, called the $\lambda$-invariant, to show, e.g. that $\Sigma^7_{1, 2}$ is not diffeomorphic to $S^7$. A little later Kervaire and Milnor {{cite|Kervaire&Milnor1963}} proved that $\Theta_7 \cong \Zz_{28}$ and Eells and Kuiper {{cite|Eells&Kuiper1962}} defined a refinement of the $\lambda$-invariant, now called the Eells-Kuiper $\mu$-invariant, which in particular gives<br />
$$ \Sigma^7_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \in bP_8 \cong \Theta_7.$$<br />
<br />
Shimada {{cite|Shimada1957}} used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres. In this case $\pi_7(SO(8)) \cong \Zz \oplus \Zz$ and the bundle $(m, n)$ has Euler number $n$ and second Pontrjagin class $6(n+2m)$. Moreover $\Theta_{15} \cong \Zz_{8,128} \oplus \Zz_2$ where the $\Zz_{8,128}$-summand is $bP_{16}$ as explained below. Results of {{cite|Wall1962a}} and {{cite|Eells&Kuiper1962}} combine to show that<br />
$$ \Sigma^{15}_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.$$<br />
<br />
*By Adams' solution of the [[Wikipedia:Hopf_invariant| Hopf-invariant]] 1 problem, {{cite|Adams1958}} and {{cite|Adams1960}}, the dimensions n = 3, 7 and 15 are the only dimensions in which a topological n-sphere can be fibre over an m-sphere for 0 < m < n. <br />
</wikitex><br />
<br />
=== Twisting ===<br />
<wikitex>;<br />
By {{cite|Cerf1970}} and {{cite|Smale1962a}} there is an isomorphism $\Theta_{n+1} \cong \Gamma_{n+1}$ for $n \geq 5$ where $\Gamma_{n+1} = \pi_0(\Diff_+(S^n))$ is the group of isotopy classes of orientation preserving diffeomorphisms of $S^n$. The map is given by<br />
$$ \Gamma_{n+1} \to \Theta_{n+1}, ~~~~~[f] \longmapsto \Sigma_{f} := D^{n+1} \cup_f (-D^{n+1}).$$<br />
Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of $S^n$ which are not isotopic to the identity. We give such a construction which probably goes back to Milnor: so far the earliest reference found is the problem list of the 1963 Seattle topology conference {{cite|Lashof1965}}.<br />
<br />
Represent $\alpha \in \pi_p(SO(q))$ and $\beta \in \pi_q(SO(p))$ by smooth compactly supported functions $\alpha : \Rr^p \to SO(q)$ and $\beta : \Rr^q \to SO(p)$ and define the following self-diffeomorphisms of $\Rr^p \times \Rr^q$<br />
$$ F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),$$<br />
$$ F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),$$<br />
$$s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.$$<br />
If follows that $s(\alpha, \beta)$ is compactly supported and so extends uniquely to a diffeomorphism of $S^{p+q}$. In this way we obtain a bilinear pairing<br />
$$ \sigma : \pi_q(SO(q)) \otimes \pi_q(SO(p)) \longrightarrow \Theta_{p+q+1}, ~~~~(\alpha, \beta) \longmapsto \Sigma_{s(\alpha, \beta)}$$<br />
such that<br />
$$ \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta).$$<br />
In particular for $k=1, 2$ we see that $\sigma(\gamma_{4k-1}', \gamma_{4k-1}') \cong -\Sigma_M$ generates $bP_{4k}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
Finding invariants of exotic sphere $\Sigma$ which distinguish it from the standard sphere is rather a subtle undertaking. Moreover such invariants are often defined via a manifold $W$ with $\partial W \cong \Sigma$. In this case finding an intrinsic definition and or computation of the relevant invariant can also be subtle.<br />
<br />
We begin by listing some invariants which are equal for all exotic spheres.<br />
{{beginthm|Proposition}} <br />
Let $\Sigma$ be a closed smooth manifold homeomorphic to the n-sphere. Then<br />
# there is an isomorphism of tangent bundles $T\Sigma \cong TS^n$,<br />
# the signature of $\Sigma$ vanishes,<br />
# the Kervaire invariant of $(\Sigma, F)$ is zero for every framing of $\Sigma$.<br />
(To make sense of the first statement remember that the topological space underlying every exotic sphere is homeomorphic to $S^n$.)<br />
{{endthm}}<br />
{{beginrem|Remark}}<br />
The analogue of the first statement for the stable tangent bundle was proven in \cite{Kervaire&Milnor1963|Theorem 3.1}. A proof of the unstable statement is given in \cite{Ray&Pedersen1980|Lemma 1.1}. The next two statements are obvious since both the signature and Kervaire invariant are defined to be zero if $n = 2k+1$ and via a symmetric or quadratic form on $H_k(\Sigma; \Zz) = 0 $ if $n = 2k$.<br />
{{endrem}} <br />
</wikitex><br />
=== Bordism classes ===<br />
<wikitex>;<br />
As every homotopy sphere is stably parallelisable, homotopy spheres admit [[B-Bordism|$B$-structures]] for any $B$. If $B$ is such that $[S^n, F] \mapsto 0 \in \Omega_n^B$ for any stable framing $F$ of $S^n$, then we obtain a well-defined homomorphism<br />
$$ \eta^B : \Theta_n \longrightarrow \Omega_n^B, ~~~\Sigma \longmapsto [\Sigma, F].$$ <br />
* If $B = BO\langle k \rangle$ for $[n/2] + 1 < k < n+2$ then $\Omega_n^B$ is isomorphic to almost framed bordism and the homomorphism $\eta^B$ is the same thing as the $\eta: \Theta_n \to \pi_n(G/O)$ in Theorem \ref{thm-ses}.<br />
* Perhaps surprisingly $\eta_n^{\Spin} \neq 0$ for all $n = 8k+1, 8k+2$, as explained in the next subsection.<br />
* In general determining $\eta^B$ is a hard an interesting problem. <br />
* $B$-coboundaries for elements of $Ker(\eta^B_n)$ are often used to define invariants of $B$-null bordant homotopy spheres.<br />
</wikitex><br />
<br />
=== The α-invariant ===<br />
<wikitex>;<br />
In dimensions $n > 1$, every exotic sphere $\Sigma$ has a unique Spin structure and from above we have the homomorphism $\eta_n^{\Spin} : \Theta_n \to \Omega_n^{\Spin}$. Recall the $\alpha$-invariant homomorphism $\alpha : \Omega_*^{\Spin} \to KO^{-*}$ and that there are isomorphisms $KO^{-8k-1} \cong KO^{-8k-2} \cong \Zz/2$ for all $k \geq 1$.<br />
{{beginthm|Theorem|\cite{Anderson&Brown&Peterson1967}}}<br />
We have $\eta_n^{\Spin}(\Sigma) = 0$ if and only if $\alpha \circ \eta_n^{\Spin}(\Sigma) = 0$ and $\eta_n^{\Spin} \neq 0$ if and only if $n = 8k+1$ or $8k+2$. <br />
{{endthm}}<br />
{{beginrem|Remark}}<br />
Exotic spheres $\Sigma$ with $\alpha(\Sigma) \neq 0$ are often called Hitchin spheres, after \cite{Hitchin1974}: see the discussion of curvature [[#Curvature on exotic spheres|below]].<br />
{{endrem}}<br />
</wikitex><br />
<br />
=== The Eells-Kuiper invariant === <br />
<wikitex>;<br />
</wikitex><br />
=== The s-invariant ===<br />
<wikitex>;<br />
</wikitex><br />
<br />
== Classification ==<br />
<wikitex>;<br />
For $n =1, 2$ and $3$, $\Theta_n = \{ S^n \}$. For $n = 4$, $\Theta_4$ is unknown. We therefore concentrate on higher dimensions.<br />
<br />
For $n \geq 5$, the group of exotic n-spheres $\Theta_n$ fits into the following long exact sequence, first discovered in {{cite|Kervaire&Milnor1963}} (more details can also be found in {{cite|Levine1983}} and {{cite|Lück2001}}):<br />
$$ \dots \stackrel{\eta_{n+1}}{\longrightarrow} \pi_{n+1}(G/O) \stackrel{\sigma_{n+1}}{\longrightarrow} L_{n+1}(e) \stackrel{\omega_{n+1}}{\longrightarrow} \Theta_n \stackrel{\eta_n}{\longrightarrow} \pi_n(G/O) \stackrel{\sigma_n}{\longrightarrow} L_n(e) \to \dots~.$$<br />
Here $L_n(e)$ is the n-th [[Wikipedia:L-theory|L-group]] of the the trivial group: $L_n(e) = \Zz, 0, \Zz/2, 0$ as n = 0, 1, 2 or 3 modulo 4 and the sequence ends at $L_5(e) = 0$. Also $O$ is the stable orthogonal group and $G$ is the stable group of homtopy self-equivalences of the sphere. There is a fibration $O \to G \to G/O$ and the groups $\pi_n(G/O)$ fit into the homtopy long exact sequence<br />
$$ \dots \to \pi_n(O) \to \pi_n(G) \to \pi_n(G/O) \to \pi_{n-1}(O) \to \pi_{n-1}(G) \to \dots $$<br />
of this fibration. The homomorphism $J_n: \pi_n(O) \to \pi_n^S \cong \pi_n(G)$ is the [[Wikipedia:J-homomorphism|stable J-homomorphism]]. In particular, by {{cite|Serre1951}} the groups $\pi_i(G)$ are finite and by {{cite|Bott1959}}, {{cite|Adams1966}} and {{cite|Quillen1971}} the domain, image and kernel of $J_n$ have been completely determined. An important result in {{cite|Kervaire&Milnor1963}} is that the homomorphism $\sigma_{4k}$ is nonzero. The above sequence then gives<br />
<br />
{{beginthm|Theorem|{{cite|Kervaire&Milnor1963}}}}\label{thm-ses}<br />
For $n \geq 5$, the group $\Theta_n$ is finite. Moreover there is an exact sequence<br />
$$ 0 \longrightarrow bP_{n+1} \longrightarrow \Theta_{n} \longrightarrow Coker(J_n) \stackrel{K_n}{\longrightarrow} C_n \longrightarrow 0 $$<br />
where $bP_{n+1} := {Im}(\omega_{n+1})$, the group of homotopy spheres bounding paralellisable manifolds, is a finite cyclic group which vanishes if $n$ is even. Moreover $C_n = 0$ unless $n = 4k+2$ when it is $0$ or $\Zz/2$.<br />
{{endthm}}<br />
<br />
The groups $Coker(J_n)$ are known for $n$ up to approximately 62. In general their determination is a very hard problem. Modulo this problem we see two remaining problems in the determination of $\Theta_n$: an extension problem and the comptutation of the order of the groups $bP_{n+1}$ and $C_n$. We discuss these in turn.<br />
<br />
{{beginthm|Theorem|{{cite|Brumfiel1968}}, {{cite|Brumfiel1969}}, {{cite|Brumfiel1970}}}}<br />
If $n \neq 2^{j} - 3$ the Kervaire-Milnor extension splits: <br />
$$\Theta_n \cong bP_{n+1} \oplus Ker(K_n).$$<br />
{{endthm}}<br />
<br />
The map $K_{4k+2} : Coker(J_{4k+2}) \to \Zz/2$ is the Kervaire invariant and by definition $C_{4k+2} = Im(K_{4k+2})$. By the long exact sequence above we have<br />
<br />
{{beginthm|Theorem|{{cite|Kervaire&Milnor1963|Section 8}}}}<br />
The group $bP_{4k+2}$ is either $\Zz/2$ or $0$. Moreover the following are equivalent:<br />
* $bP_{4k+2} = 0$,<br />
* the Kervaire sphere $\Sigma^{4k+1}_K$ is diffeomorphic to the standard sphere,<br />
* there is a framed manifold with Kervaire invariant 1: $C_{4k+2} \cong \Zz/2$.<br />
Conversely the following are equivalent:<br />
* $bP_{4k+2} = \Zz/2$,<br />
* the Kervaire sphere $\Sigma^{4k+1}_K$ is not diffeomorphic to the standard sphere,<br />
* there is no framed manifold with Kervaire invariant 1: $C_{4k+2} \cong 0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== The orders of bP<sub>4k</sub> and bP<sub>4k+2</sub> ===<br />
<wikitex>;<br />
The group $bP_{4k}$ is a cyclic group whose order can be determined using the Hirzebruch's signature theorem if one knows the order of $Im(J_{4k-1}) \subset \pi_{4k-1}(G)$. Adams determined the latter group up to a factor of two which was settled by Quillen with a positive solution to the Adams conjecture. <br />
<br />
{{beginthm|Theorem}}<br />
Let $a_k = (3-(-1)^k)/2$, let $B_k$ be the k-th Bernoulli number (topologist indexing) and for $x \in \Qq$ let $Num(x)$ denote the numerator of $x$ expressed in lowest form. Then for $k \geq 2$, the order of $bP_{4k}$ is<br />
$$ t_k = a_k \cdot 2^{2k-2}\cdot (2^{2k-1}-1) \cdot Num(B_k/4k).$$<br />
{{endthm}}<br />
<br />
{{beginrem|Remark}}<br />
Note that $Num(B_k/4k)$ is odd so the 2-primary order of $bP_{4k}$ is $a_k \cdot 2^{2k-2}$ while the odd part is $(2^{2k-1}-1) \cdot Num(B_k/4k)$. Modulo the Adams conjecture the proof appeared in {{cite|Kervaire&Milnor1963|Section 7}}. Detailed treatments can also be found in {{cite|Levine1983|Section 3}} and {{cite|Lück2001|Chapter 6}}.<br />
{{endrem}}<br />
<br />
The next theorem describes the situation for $bP_{4k+2}$ which is now almost completely understood as well. References for the theorem are given in the remark which follows it.<br />
<br />
{{beginthm|Theorem}}<br />
The group $bP_{4k+2}$ is given as follows:<br />
* $bP_{6} = bP_{14} = bP_{30} = bP_{62} = 0$,<br />
* $bP_{126} = 0$ or $\Zz/2$,<br />
* $bP_{4k+2} = \Zz/2$ else.<br />
{{endthm}}<br />
<br />
{{beginrem|Remark}}<br />
The following is a chronological list of determinations of $bP_{4k+2}$:<br />
* $bP_{10} = \Zz/2$, {{cite|Kervaire1960a}}.<br />
* $bP_{6} = bP_{14} = 0$ {{cite|Kervaire&Milnor1963}}.<br />
* $bP_{8k+2} = \Zz/2$, {{cite|Anderson&Brown&Peterson1966a}}.<br />
* $bP_{30} = 0$, {{cite|Mahowald&Tangora1967}}.<br />
* $bP_{4k+2} = \Zz/2$ unless $4k+2 = 2^j - 2$ {{cite|Browder1969}}.<br />
* $bP_{62} = 0$, {{cite|Barratt&Jones&Mahowald1984}}.<br />
* $bP_{2^j - 2} = \Zz/2$ for $j \geq 8$, {{cite|Hill&Hopkins&Ravenel2009}}.<br />
{{endrem}}<br />
</wikitex><br />
<br />
== Further discussion ==<br />
=== Curvature on exotic spheres ===<br />
<wikitex>;<br />
Gromoll-Meyer proved that a certain exotic 7-sphere can be realized as a biquotient of the compact Lie group Sp(2) and thus has a Riemannian metric of nonnegative sectional curvature. It is not known whether there exist exotic spheres with Riemannian metrics of positive sectional curvature. For a recent review of which exotic spheres admit metrics of various sorts of positive curvature see \cite{Joachim&Wraith2008}.<br />
</wikitex><br />
<br />
=== The Kervaire-Milnor braid ===<br />
<wikitex>;<br />
$$<br />
\def\curv{1.5pc}% Adjust the curvature of the curved arrows here<br />
\xymatrix@!R@!C@!0@R=2.5pc@C=4pc{% Adjust the spacing here<br />
\pi_n(\Top/O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(G) \\<br />
& \pi_n(G/O) \ar[dr] \ar[ur] && \pi_{n-1}(\Top) \ar[dr] \ar[ur] \\<br />
\pi_n(G) \ar[ur] \ar@/d\curv/[rr] && \pi_n(G/\Top) \ar[ur] \ar@/d\curv/[rr] && \pi_{n-1}(\Top/O)<br />
}<br />
$$<br />
</wikitex><br />
== PL manifolds admitting no smooth structure ==<br />
<wikitex>;<br />
Let $W^{2n}$ be a [[Exotic spheres#Plumbing|plumbing manifold]] as described above. By a simple version of the [[Alexander trick]], there is a homemorphism $f \colon \partial W \cong S^{2n-1}$ and so we can form the closed topological manifold<br />
$$ \bar W : = W \cup_f D^{2n}.$$<br />
If $\partial W$ is exotic then it turns out that $\bar W$ is a topological manifold which admits no smooth structure! <br />
<br />
\cite{Kervaire1960a} shows that $\bar W^{10}$ is non-smoothable and the arugments there work for all odd $n$ so long as the Kervaire sphere is exotic. <br />
<br />
When $n$ is even the proof is more complicated: one first need's Novikov's theorem that the rational Pontrjagin classes of a topological manifold are homeomorphism invariants \cite{Novikov1965b}. Prior to Novikvo's result, some weaker statements were known. For example, when $n=4$ and $W$ is the total space of a [[Exotic spheres#Sphere bundles|$D^4$-bundle]] over $S^4$ as above and if $\partial W = \Sigma_{m, 1}$ then by \cite{Tamura1961} $\bar W$ is smoothable if and only if $m(m-1)/2 \equiv 0$ mod $4$.</wikitex><ref>Note that Tamura uses a different identification <tex>\pi_3(SO(4)) \cong \Zz \oplus \Zz</tex> from the one used above.</ref><wikitex> Applying Novikov's theorem we know that $\bar W$ is smoothable if and only if $m(m-1)/2 \equiv 0$ mod $56$.<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
== Footnotes ==<br />
<references/><br />
<br />
== External links ==<br />
* The Wikipedia page on [[Wikipedia:Exotic sphere|exotic spheres]]<br />
* The tabulation of the order of the group of exotic spheres in the [http://oeis.org/classic/A001676 On-Line Encyclopedia of Integer Sequences]<br />
* Andrew Ranicki's exotic sphere home page, with many of the original papers: [http://www.maths.ed.ac.uk/~aar/exotic.htm http://www.maths.ed.ac.uk/~aar/exotic.htm] <br />
** Including some [http://www.maths.ed.ac.uk/~aar/papers/km-it.pdf original correspondence between Kervaire and Milnor]<br />
*[http://www.nilesjohnson.net/seven-manifolds.html An animation of exotic 7-spheres]. Slides from a presentation by [http://www.nilesjohnson.net/ Nile Johsnon] at the [http://www.ima.umn.edu/2011-2012/SW1.30-2.1.12/ Second Abel conference] in honor of [[Wikipedia:John Milnor|John Milnor]].<br />
[[Category:Manifolds]]<br />
[[Category:Highly-connected manifolds]]<br />
[[Category:Surgery]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Exotic_spheresExotic spheres2012-03-23T01:58:49Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
By a homotopy sphere $\Sigma^n$ we mean a closed smooth oriented n-manifold homotopy equivalent to $S^n$. The manifold $\Sigma^n$ is called an exotic sphere if it is not diffeomorphic to $S^n$. By the Generalised Poincaré Conjecture proven by Smale, every homotopy sphere in dimension $n \geq 5$ is homeomorphic to $S^n$: this statement holds in dimension 2 by the classification of [[Surfaces|surfaces]] and was famously proven in dimension 4 in {{cite|Freedman1982}} and in dimension 3 by Perelman. We define <br />
$$\Theta_{n} := \{[\Sigma^n] | \Sigma^n \simeq S^n \}$$<br />
to be the set of oriented diffeomorphism classes of homotopy spheres. [[Wikipedia:Connected_sum|Connected sum]] makes $\Theta_n$ into an abelian group with inverse given by reversing orientation. An important subgroup of $\Theta_n$ is $bP_{n+1}$ which consists of those homotopy spheres which bound parallelisable manifolds.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
The first exotic spheres discovered were certain 3-sphere bundles over the 4-sphere, {{cite|Milnor1956}}. Following this discovery there was a rapid development of techniques which construct exotic spheres. We review four such constructions: plumbing, Brieskorn varieties, sphere-bundles and twisting. <br />
</wikitex><br />
<br />
=== Plumbing ===<br />
<wikitex>;<br />
As special case of the following construction goes back at least to {{cite|Milnor1959}}.<br />
<br />
Let $i \in \{1, \dots, n\}$, let $(p_i, q_i)$ be pairs of positive integers such that $p_i + q_i + 2 = n$ and let $\alpha_i \in \pi_{p_i}(SO(q_i+1))$ be the clutching functions of $D^{q_i+1}$-bundles over $S^{p_i + 1}$<br />
$$ D^{q_i+1} \to D(\alpha_i) \to S^{p_i+1}.$$<br />
Let $G$ be a graph with vertices $\{v_1, \dots, v_n\}$ such that the edge set between $v_i$ and $v_j$, is non-empty only if $p_i = q_j$. We form the manifold $W = W(G;\{\alpha_i\})$ from the disjoint union of the $D(\alpha_i)$ by identifying $D^{p_i+1} \times D^{q_i+1}$ and $D^{q_j+1} \times D^{p_j+1}$ for each edge in $G$. If $G$ is simply connected then <br />
$$\Sigma(G, \{\alpha_i \}) : = \partial W$$<br />
is often a homotopy sphere. We establish some notation for graphs, bundles and define <br />
* let $T$ denote the graph with two vertices and one edge connecting them and define $\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$,<br />
* let $E_8$ denote the $E_8$-graph,<br />
* let $\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$-sphere,<br />
* let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$, $k > 4s$, denote a generator,<br />
* let $\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$, denote a generator: <br />
* let $S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$ be the suspension homomorphism,<br />
**$S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$ for $k = 1, 2$ and $S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$ for $k > 2$,<br />
* let $\eta_n : S^{n+1} \to S^n$ be essential.<br />
<br />
Then we have the following exotic spheres.<br />
<br />
* $\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$, the Milnor sphere, generates $bP_{4k}$, $k>1$.<br />
* $\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$, the Kervaire sphere, generates $bP_{4k+2}$.<br />
* $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$. <br />
**For general $k$, $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic.<br />
* $\Sigma^8(\gamma_3^5, \eta_3\tau_4)$, generates $\Theta_8 = \Zz_2$. <br />
* $\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$, generates $\Theta_{16} = \Zz_2$. <br />
</wikitex><br />
<br />
=== Brieskorn varieties ===<br />
<wikitex>;<br />
Let $z = (z_0, \dots , z_n)$ be a point in $\Cc^{n+1}$ and let $a = (a_0, \dots, a_n)$ be a string of n+1 positive integers. Given the complex variety $V(a) : = \{z \, | \, z_0^{a_0} + \dots + z_n^{a_n} =0 \}$ and the $\epsilon$-sphere $S^{2n+1}_\epsilon : = \{ z \, | \, \Sigma_{i=0}^n z_i\bar z_i = \epsilon \}$ for small $\epsilon$, following {{cite|Milnor1968}}<br />
we define the closed smooth oriented $(n-2)$-connected $(2n-1)$-manifold<br />
$$ W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.$$<br />
The manifolds $W^{2n-1}(a)$ are often called Brieskorn varieties. By construction, every $W^{2n-1}(a)$ lies in $S^{2n+1}$ and so bounds a parallelisable manifold. In {{cite|Brieskorn1966}} and {{cite|Brieskorn1966a}} (see also {{cite|Hirzebruch&Mayer1968}}), it is shown in particular that all homotopy spheres in $bP_{4k}$ and $bP_{4k-2}$ can be realised as $W(a)$ for some $a$. Let $2, \dots, 2$ be a string of 2k-1 2's in a row with $k \geq 2$, then there are diffeomorphisms<br />
$$ W^{4k-1}(3, 6r-1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},$$<br />
$$ W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.$$<br />
</wikitex><br />
<br />
=== Sphere bundles ===<br />
<wikitex>;<br />
The first known examples of exotic spheres were discovered by Milnor in {{cite|Milnor1956}}. They are the total spaces of certain 3-[[Wikipedia:Sphere_bundle#Sphere_bundles|sphere bundles]] over the 4-sphere as we now explain: the group $\pi_3(SO(4)) \cong \Zz \oplus \Zz$ parametrises linear $3$-sphere bundles over $S^4$ where a pair $(m, n)$ gives rise to a bundle with Euler number $n$ and first Pontrjagin class $2(n+2m)$: here we orient $S^4$ and so identify $H^4(S^4; \Zz) = \Zz$. If we set $n = 1$ then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold $\Sigma^7_{m, 1}$, the total space of the bundle $(m, 1)$, is a homotopy sphere. Milnor first used a $\Zz_7$-invariant, called the $\lambda$-invariant, to show, e.g. that $\Sigma^7_{1, 2}$ is not diffeomorphic to $S^7$. A little later Kervaire and Milnor {{cite|Kervaire&Milnor1963}} proved that $\Theta_7 \cong \Zz_{28}$ and Eells and Kuiper {{cite|Eells&Kuiper1962}} defined a refinement of the $\lambda$-invariant, now called the Eells-Kuiper $\mu$-invariant, which in particular gives<br />
$$ \Sigma^7_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \in bP_8 \cong \Theta_7.$$<br />
<br />
Shimada {{cite|Shimada1957}} used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres. In this case $\pi_7(SO(8)) \cong \Zz \oplus \Zz$ and the bundle $(m, n)$ has Euler number $n$ and second Pontrjagin class $6(n+2m)$. Moreover $\Theta_{15} \cong \Zz_{8,128} \oplus \Zz_2$ where the $\Zz_{8,128}$-summand is $bP_{16}$ as explained below. Results of {{cite|Wall1962a}} and {{cite|Eells&Kuiper1962}} combine to show that<br />
$$ \Sigma^{15}_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.$$<br />
<br />
*By Adams' solution of the [[Wikipedia:Hopf_invariant| Hopf-invariant]] 1 problem, {{cite|Adams1958}} and {{cite|Adams1960}}, the dimensions n = 3, 7 and 15 are the only dimensions in which a topological n-sphere can be fibre over an m-sphere for 0 < m < n. <br />
</wikitex><br />
<br />
=== Twisting ===<br />
<wikitex>;<br />
By {{cite|Cerf1970}} and {{cite|Smale1962a}} there is an isomorphism $\Theta_{n+1} \cong \Gamma_{n+1}$ for $n \geq 5$ where $\Gamma_{n+1} = \pi_0(\Diff_+(S^n))$ is the group of isotopy classes of orientation preserving diffeomorphisms of $S^n$. The map is given by<br />
$$ \Gamma_{n+1} \to \Theta_{n+1}, ~~~~~[f] \longmapsto \Sigma_{f} := D^{n+1} \cup_f (-D^{n+1}).$$<br />
Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of $S^n$ which are not isotopic to the identity. We give such a construction which probably goes back to Milnor: so far the earliest reference found is the problem list of the 1963 Seattle topology conference {{cite|Lashof1965}}.<br />
<br />
Represent $\alpha \in \pi_p(SO(q))$ and $\beta \in \pi_q(SO(p))$ by smooth compactly supported functions $\alpha : \Rr^p \to SO(q)$ and $\beta : \Rr^q \to SO(p)$ and define the following self-diffeomorphisms of $\Rr^p \times \Rr^q$<br />
$$ F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),$$<br />
$$ F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),$$<br />
$$s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.$$<br />
If follows that $s(\alpha, \beta)$ is compactly supported and so extends uniquely to a diffeomorphism of $S^{p+q}$. In this way we obtain a bilinear pairing<br />
$$ \sigma : \pi_q(SO(q)) \otimes \pi_q(SO(p)) \longrightarrow \Theta_{p+q+1}, ~~~~(\alpha, \beta) \longmapsto \Sigma_{s(\alpha, \beta)}$$<br />
such that<br />
$$ \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta).$$<br />
In particular for $k=1, 2$ we see that $\sigma(\gamma_{4k-1}', \gamma_{4k-1}') \cong -\Sigma_M$ generates $bP_{4k}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
Finding invariants of exotic sphere $\Sigma$ which distinguish it from the standard sphere is rather a subtle undertaking. Moreover such invariants are often defined via a manifold $W$ with $\partial W \cong \Sigma$. In this case finding an intrinsic definition and or computation of the relevant invariant can also be subtle.<br />
<br />
We begin by listing some invariants which are equal for all exotic spheres.<br />
{{beginthm|Proposition}} <br />
Let $\Sigma$ be a closed smooth manifold homeomorphic to the n-sphere. Then<br />
# there is an isomorphism of tangent bundles $T\Sigma \cong TS^n$,<br />
# the signature of $\Sigma$ vanishes,<br />
# the Kervaire invariant of $(\Sigma, F)$ is zero for every framing of $\Sigma$.<br />
(To make sense of the first statement remember that the topological space underlying every exotic sphere is homeomorphic to $S^n$.)<br />
{{endthm}}<br />
{{beginrem|Remark}}<br />
The analogue of the first statement for the stable tangent bundle was proven in \cite{Kervaire&Milnor1963|Theorem 3.1}. A proof of the unstable statement is given in \cite{Ray&Pedersen1980|Lemma 1.1}. The next two statements are obvious since both the signature and Kervaire invariant are defined to be zero if $n = 2k+1$ and via a symmetric or quadratic form on $H_k(\Sigma; \Zz) = 0 $ if $n = 2k$.<br />
{{endrem}} <br />
</wikitex><br />
=== Bordism classes ===<br />
<wikitex>;<br />
As every homotopy sphere is stably parallelisable, homotopy spheres admit [[B-Bordism|$B$-structures]] for any $B$. If $B$ is such that $[S^n, F] \mapsto 0 \in \Omega_n^B$ for any stable framing $F$ of $S^n$, then we obtain a well-defined homomorphism<br />
$$ \eta^B : \Theta_n \longrightarrow \Omega_n^B, ~~~\Sigma \longmapsto [\Sigma, F].$$ <br />
* If $B = BO\langle k \rangle$ for $[n/2] + 1 < k < n+2$ then $\Omega_n^B$ is isomorphic to almost framed bordism and the homomorphism $\eta^B$ is the same thing as the $\eta: \Theta_n \to \pi_n(G/O)$ in Theorem \ref{thm-ses}.<br />
* Perhaps surprisingly $\eta_n^{\Spin} \neq 0$ for all $n = 8k+1, 8k+2$, as explained in the next subsection.<br />
* In general determining $\eta^B$ is a hard an interesting problem. <br />
* $B$-coboundaries for elements of $Ker(\eta^B_n)$ are often used to define invariants of $B$-null bordant homotopy spheres.<br />
</wikitex><br />
<br />
=== The α-invariant ===<br />
<wikitex>;<br />
In dimensions $n > 1$, every exotic sphere $\Sigma$ has a unique Spin structure and from above we have the homomorphism $\eta_n^{\Spin} : \Theta_n \to \Omega_n^{\Spin}$. Recall the $\alpha$-invariant homomorphism $\alpha : \Omega_*^{\Spin} \to KO^{-*}$ and that there are isomorphisms $KO^{-8k-1} \cong KO^{-8k-2} \cong \Zz/2$ for all $k \geq 1$.<br />
{{beginthm|Theorem|\cite{Anderson&Brown&Peterson1967}}}<br />
We have $\eta_n^{\Spin}(\Sigma) = 0$ if and only if $\alpha \circ \eta_n^{\Spin}(\Sigma) = 0$ and $\eta_n^{\Spin} \neq 0$ if and only if $n = 8k+1$ or $8k+2$. <br />
{{endthm}}<br />
{{beginrem|Remark}}<br />
Exotic spheres $\Sigma$ with $\alpha(\Sigma) \neq 0$ are often called Hitchin spheres, after \cite{Hitchin1974}: see the discussion of curvature [[#Curvature on exotic spheres|below]].<br />
{{endrem}}<br />
</wikitex><br />
<br />
=== The Eells-Kuiper invariant === <br />
<wikitex>;<br />
</wikitex><br />
=== The s-invariant ===<br />
<wikitex>;<br />
</wikitex><br />
<br />
== Classification ==<br />
<wikitex>;<br />
For $n =1, 2$ and $3$, $\Theta_n = \{ S^n \}$. For $n = 4$, $\Theta_4$ is unknown. We therefore concentrate on higher dimensions.<br />
<br />
For $n \geq 5$, the group of exotic n-spheres $\Theta_n$ fits into the following long exact sequence, first discovered in {{cite|Kervaire&Milnor1963}} (more details can also be found in {{cite|Levine1983}} and {{cite|Lück2001}}):<br />
$$ \dots \stackrel{\eta_{n+1}}{\longrightarrow} \pi_{n+1}(G/O) \stackrel{\sigma_{n+1}}{\longrightarrow} L_{n+1}(e) \stackrel{\omega_{n+1}}{\longrightarrow} \Theta_n \stackrel{\eta_n}{\longrightarrow} \pi_n(G/O) \stackrel{\sigma_n}{\longrightarrow} L_n(e) \to \dots~.$$<br />
Here $L_n(e)$ is the n-th [[Wikipedia:L-theory|L-group]] of the the trivial group: $L_n(e) = \Zz, 0, \Zz/2, 0$ as n = 0, 1, 2 or 3 modulo 4 and the sequence ends at $L_5(e) = 0$. Also $O$ is the stable orthogonal group and $G$ is the stable group of homtopy self-equivalences of the sphere. There is a fibration $O \to G \to G/O$ and the groups $\pi_n(G/O)$ fit into the homtopy long exact sequence<br />
$$ \dots \to \pi_n(O) \to \pi_n(G) \to \pi_n(G/O) \to \pi_{n-1}(O) \to \pi_{n-1}(G) \to \dots $$<br />
of this fibration. The homomorphism $J_n: \pi_n(O) \to \pi_n^S \cong \pi_n(G)$ is the [[Wikipedia:J-homomorphism|stable J-homomorphism]]. In particular, by {{cite|Serre1951}} the groups $\pi_i(G)$ are finite and by {{cite|Bott1959}}, {{cite|Adams1966}} and {{cite|Quillen1971}} the domain, image and kernel of $J_n$ have been completely determined. An important result in {{cite|Kervaire&Milnor1963}} is that the homomorphism $\sigma_{4k}$ is nonzero. The above sequence then gives<br />
<br />
{{beginthm|Theorem|{{cite|Kervaire&Milnor1963}}}}\label{thm-ses}<br />
For $n \geq 5$, the group $\Theta_n$ is finite. Moreover there is an exact sequence<br />
$$ 0 \longrightarrow bP_{n+1} \longrightarrow \Theta_{n} \longrightarrow Coker(J_n) \stackrel{K_n}{\longrightarrow} C_n \longrightarrow 0 $$<br />
where $bP_{n+1} := {Im}(\omega_{n+1})$, the group of homotopy spheres bounding paralellisable manifolds, is a finite cyclic group which vanishes if $n$ is even. Moreover $C_n = 0$ unless $n = 4k+2$ when it is $0$ or $\Zz/2$.<br />
{{endthm}}<br />
<br />
The groups $Coker(J_n)$ are known for $n$ up to approximately 62. In general their determination is a very hard problem. Modulo this problem we see two remaining problems in the determination of $\Theta_n$: an extension problem and the comptutation of the order of the groups $bP_{n+1}$ and $C_n$. We discuss these in turn.<br />
<br />
{{beginthm|Theorem|{{cite|Brumfiel1968}}, {{cite|Brumfiel1969}}, {{cite|Brumfiel1970}}}}<br />
If $n \neq 2^{j} - 3$ the Kervaire-Milnor extension splits: <br />
$$\Theta_n \cong bP_{n+1} \oplus Ker(K_n).$$<br />
{{endthm}}<br />
<br />
The map $K_{4k+2} : Coker(J_{4k+2}) \to \Zz/2$ is the Kervaire invariant and by definition $C_{4k+2} = Im(K_{4k+2})$. By the long exact sequence above we have<br />
<br />
{{beginthm|Theorem|{{cite|Kervaire&Milnor1963|Section 8}}}}<br />
The group $bP_{4k+2}$ is either $\Zz/2$ or $0$. Moreover the following are equivalent:<br />
* $bP_{4k+2} = 0$,<br />
* the Kervaire sphere $\Sigma^{4k+1}_K$ is diffeomorphic to the standard sphere,<br />
* there is a framed manifold with Kervaire invariant 1: $C_{4k+2} \cong \Zz/2$.<br />
Conversely the following are equivalent:<br />
* $bP_{4k+2} = \Zz/2$,<br />
* the Kervaire sphere $\Sigma^{4k+1}_K$ is not diffeomorphic to the standard sphere,<br />
* there is no framed manifold with Kervaire invariant 1: $C_{4k+2} \cong 0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== The orders of bP<sub>4k</sub> and bP<sub>4k+2</sub> ===<br />
<wikitex>;<br />
The group $bP_{4k}$ is a cyclic group whose order can be determined using the Hirzebruch's signature theorem if one knows the order of $Im(J_{4k-1}) \subset \pi_{4k-1}(G)$. Adams determined the latter group up to a factor of two which was settled by Quillen with a positive solution to the Adams conjecture. <br />
<br />
{{beginthm|Theorem}}<br />
Let $a_k = (3-(-1)^k)/2$, let $B_k$ be the k-th Bernoulli number (topologist indexing) and for $x \in \Qq$ let $Num(x)$ denote the numerator of $x$ expressed in lowest form. Then for $k \geq 2$, the order of $bP_{4k}$ is<br />
$$ t_k = a_k \cdot 2^{2k-2}\cdot (2^{2k-1}-1) \cdot Num(B_k/4k).$$<br />
{{endthm}}<br />
<br />
{{beginrem|Remark}}<br />
Note that $Num(B_k/4k)$ is odd so the 2-primary order of $bP_{4k}$ is $a_k \cdot 2^{2k-2}$ while the odd part is $(2^{2k-1}-1) \cdot Num(B_k/4k)$. Modulo the Adams conjecture the proof appeared in {{cite|Kervaire&Milnor1963|Section 7}}. Detailed treatments can also be found in {{cite|Levine1983|Section 3}} and {{cite|Lück2001|Chapter 6}}.<br />
{{endrem}}<br />
<br />
The next theorem describes the situation for $bP_{4k+2}$ which is now almost completely understood as well. References for the theorem are given in the remark which follows it.<br />
<br />
{{beginthm|Theorem}}<br />
The group $bP_{4k+2}$ is given as follows:<br />
* $bP_{6} = bP_{14} = bP_{30} = bP_{62} = 0$,<br />
* $bP_{126} = 0$ or $\Zz/2$,<br />
* $bP_{4k+2} = \Zz/2$ else.<br />
{{endthm}}<br />
<br />
{{beginrem|Remark}}<br />
The following is a chronological list of determinations of $bP_{4k+2}$:<br />
* $bP_{10} = \Zz/2$, {{cite|Kervaire1960a}}.<br />
* $bP_{6} = bP_{14} = 0$ {{cite|Kervaire&Milnor1963}}.<br />
* $bP_{8k+2} = \Zz/2$, {{cite|Anderson&Brown&Peterson1966a}}.<br />
* $bP_{30} = 0$, {{cite|Mahowald&Tangora1967}}.<br />
* $bP_{4k+2} = \Zz/2$ unless $4k+2 = 2^j - 2$ {{cite|Browder1969}}.<br />
* $bP_{62} = 0$, {{cite|Barratt&Jones&Mahowald1984}}.<br />
* $bP_{2^j - 2} = \Zz/2$ for $j \geq 8$, {{cite|Hill&Hopkins&Ravenel2009}}.<br />
{{endrem}}<br />
</wikitex><br />
<br />
== Further discussion ==<br />
=== Curvature on exotic spheres ===<br />
<wikitex>;<br />
Gromoll-Meyer proved that a certain exotic 7-sphere can be realized as a biquotient of the compact Lie group Sp(2) and has a Riemannian metric of nonnegative sectional curvature. It is not known whether there exist exotic spheres with Riemannian metrics of positive sectional curvature. For a recent review of which exotic spheres admit metrics of various sorts of positive curvature see \cite{Joachim&Wraith2008}.<br />
</wikitex><br />
<br />
=== The Kervaire-Milnor braid ===<br />
<wikitex>;<br />
$$<br />
\def\curv{1.5pc}% Adjust the curvature of the curved arrows here<br />
\xymatrix@!R@!C@!0@R=2.5pc@C=4pc{% Adjust the spacing here<br />
\pi_n(\Top/O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(G) \\<br />
& \pi_n(G/O) \ar[dr] \ar[ur] && \pi_{n-1}(\Top) \ar[dr] \ar[ur] \\<br />
\pi_n(G) \ar[ur] \ar@/d\curv/[rr] && \pi_n(G/\Top) \ar[ur] \ar@/d\curv/[rr] && \pi_{n-1}(\Top/O)<br />
}<br />
$$<br />
</wikitex><br />
== PL manifolds admitting no smooth structure ==<br />
<wikitex>;<br />
Let $W^{2n}$ be a [[Exotic spheres#Plumbing|plumbing manifold]] as described above. By a simple version of the [[Alexander trick]], there is a homemorphism $f \colon \partial W \cong S^{2n-1}$ and so we can form the closed topological manifold<br />
$$ \bar W : = W \cup_f D^{2n}.$$<br />
If $\partial W$ is exotic then it turns out that $\bar W$ is a topological manifold which admits no smooth structure! <br />
<br />
\cite{Kervaire1960a} shows that $\bar W^{10}$ is non-smoothable and the arugments there work for all odd $n$ so long as the Kervaire sphere is exotic. <br />
<br />
When $n$ is even the proof is more complicated: one first need's Novikov's theorem that the rational Pontrjagin classes of a topological manifold are homeomorphism invariants \cite{Novikov1965b}. Prior to Novikvo's result, some weaker statements were known. For example, when $n=4$ and $W$ is the total space of a [[Exotic spheres#Sphere bundles|$D^4$-bundle]] over $S^4$ as above and if $\partial W = \Sigma_{m, 1}$ then by \cite{Tamura1961} $\bar W$ is smoothable if and only if $m(m-1)/2 \equiv 0$ mod $4$.</wikitex><ref>Note that Tamura uses a different identification <tex>\pi_3(SO(4)) \cong \Zz \oplus \Zz</tex> from the one used above.</ref><wikitex> Applying Novikov's theorem we know that $\bar W$ is smoothable if and only if $m(m-1)/2 \equiv 0$ mod $56$.<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
== Footnotes ==<br />
<references/><br />
<br />
== External links ==<br />
* The Wikipedia page on [[Wikipedia:Exotic sphere|exotic spheres]]<br />
* The tabulation of the order of the group of exotic spheres in the [http://oeis.org/classic/A001676 On-Line Encyclopedia of Integer Sequences]<br />
* Andrew Ranicki's exotic sphere home page, with many of the original papers: [http://www.maths.ed.ac.uk/~aar/exotic.htm http://www.maths.ed.ac.uk/~aar/exotic.htm] <br />
** Including some [http://www.maths.ed.ac.uk/~aar/papers/km-it.pdf original correspondence between Kervaire and Milnor]<br />
*[http://www.nilesjohnson.net/seven-manifolds.html An animation of exotic 7-spheres]. Slides from a presentation by [http://www.nilesjohnson.net/ Nile Johsnon] at the [http://www.ima.umn.edu/2011-2012/SW1.30-2.1.12/ Second Abel conference] in honor of [[Wikipedia:John Milnor|John Milnor]].<br />
[[Category:Manifolds]]<br />
[[Category:Highly-connected manifolds]]<br />
[[Category:Surgery]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Exotic_spheresExotic spheres2012-03-23T01:57:49Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
By a homotopy sphere $\Sigma^n$ we mean a closed smooth oriented n-manifold homotopy equivalent to $S^n$. The manifold $\Sigma^n$ is called an exotic sphere if it is not diffeomorphic to $S^n$. By the Generalised Poincaré Conjecture proven by Smale, every homotopy sphere in dimension $n \geq 5$ is homeomorphic to $S^n$: this statement holds in dimension 2 by the classification of [[Surfaces|surfaces]] and was famously proven in dimension 4 in {{cite|Freedman1982}} and in dimension 3 by Perelman. We define <br />
$$\Theta_{n} := \{[\Sigma^n] | \Sigma^n \simeq S^n \}$$<br />
to be the set of oriented diffeomorphism classes of homotopy spheres. [[Wikipedia:Connected_sum|Connected sum]] makes $\Theta_n$ into an abelian group with inverse given by reversing orientation. An important subgroup of $\Theta_n$ is $bP_{n+1}$ which consists of those homotopy spheres which bound parallelisable manifolds.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
The first exotic spheres discovered were certain 3-sphere bundles over the 4-sphere, {{cite|Milnor1956}}. Following this discovery there was a rapid development of techniques which construct exotic spheres. We review four such constructions: plumbing, Brieskorn varieties, sphere-bundles and twisting. <br />
</wikitex><br />
<br />
=== Plumbing ===<br />
<wikitex>;<br />
As special case of the following construction goes back at least to {{cite|Milnor1959}}.<br />
<br />
Let $i \in \{1, \dots, n\}$, let $(p_i, q_i)$ be pairs of positive integers such that $p_i + q_i + 2 = n$ and let $\alpha_i \in \pi_{p_i}(SO(q_i+1))$ be the clutching functions of $D^{q_i+1}$-bundles over $S^{p_i + 1}$<br />
$$ D^{q_i+1} \to D(\alpha_i) \to S^{p_i+1}.$$<br />
Let $G$ be a graph with vertices $\{v_1, \dots, v_n\}$ such that the edge set between $v_i$ and $v_j$, is non-empty only if $p_i = q_j$. We form the manifold $W = W(G;\{\alpha_i\})$ from the disjoint union of the $D(\alpha_i)$ by identifying $D^{p_i+1} \times D^{q_i+1}$ and $D^{q_j+1} \times D^{p_j+1}$ for each edge in $G$. If $G$ is simply connected then <br />
$$\Sigma(G, \{\alpha_i \}) : = \partial W$$<br />
is often a homotopy sphere. We establish some notation for graphs, bundles and define <br />
* let $T$ denote the graph with two vertices and one edge connecting them and define $\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$,<br />
* let $E_8$ denote the $E_8$-graph,<br />
* let $\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$-sphere,<br />
* let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$, $k > 4s$, denote a generator,<br />
* let $\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$, denote a generator: <br />
* let $S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$ be the suspension homomorphism,<br />
**$S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$ for $k = 1, 2$ and $S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$ for $k > 2$,<br />
* let $\eta_n : S^{n+1} \to S^n$ be essential.<br />
<br />
Then we have the following exotic spheres.<br />
<br />
* $\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$, the Milnor sphere, generates $bP_{4k}$, $k>1$.<br />
* $\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$, the Kervaire sphere, generates $bP_{4k+2}$.<br />
* $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$. <br />
**For general $k$, $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic.<br />
* $\Sigma^8(\gamma_3^5, \eta_3\tau_4)$, generates $\Theta_8 = \Zz_2$. <br />
* $\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$, generates $\Theta_{16} = \Zz_2$. <br />
</wikitex><br />
<br />
=== Brieskorn varieties ===<br />
<wikitex>;<br />
Let $z = (z_0, \dots , z_n)$ be a point in $\Cc^{n+1}$ and let $a = (a_0, \dots, a_n)$ be a string of n+1 positive integers. Given the complex variety $V(a) : = \{z \, | \, z_0^{a_0} + \dots + z_n^{a_n} =0 \}$ and the $\epsilon$-sphere $S^{2n+1}_\epsilon : = \{ z \, | \, \Sigma_{i=0}^n z_i\bar z_i = \epsilon \}$ for small $\epsilon$, following {{cite|Milnor1968}}<br />
we define the closed smooth oriented $(n-2)$-connected $(2n-1)$-manifold<br />
$$ W^{2n-1}(a) : = V(a) \cap S^{2n+1}_\epsilon.$$<br />
The manifolds $W^{2n-1}(a)$ are often called Brieskorn varieties. By construction, every $W^{2n-1}(a)$ lies in $S^{2n+1}$ and so bounds a parallelisable manifold. In {{cite|Brieskorn1966}} and {{cite|Brieskorn1966a}} (see also {{cite|Hirzebruch&Mayer1968}}), it is shown in particular that all homotopy spheres in $bP_{4k}$ and $bP_{4k-2}$ can be realised as $W(a)$ for some $a$. Let $2, \dots, 2$ be a string of 2k-1 2's in a row with $k \geq 2$, then there are diffeomorphisms<br />
$$ W^{4k-1}(3, 6r-1, 2, \dots , 2) \cong r \cdot \Sigma_M \in bP_{4k},$$<br />
$$ W^{4k-3}(3, 2, \dots, 2) \cong \Sigma_K \in bP_{4k-2}.$$<br />
</wikitex><br />
<br />
=== Sphere bundles ===<br />
<wikitex>;<br />
The first known examples of exotic spheres were discovered by Milnor in {{cite|Milnor1956}}. They are the total spaces of certain 3-[[Wikipedia:Sphere_bundle#Sphere_bundles|sphere bundles]] over the 4-sphere as we now explain: the group $\pi_3(SO(4)) \cong \Zz \oplus \Zz$ parametrises linear $3$-sphere bundles over $S^4$ where a pair $(m, n)$ gives rise to a bundle with Euler number $n$ and first Pontrjagin class $2(n+2m)$: here we orient $S^4$ and so identify $H^4(S^4; \Zz) = \Zz$. If we set $n = 1$ then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold $\Sigma^7_{m, 1}$, the total space of the bundle $(m, 1)$, is a homotopy sphere. Milnor first used a $\Zz_7$-invariant, called the $\lambda$-invariant, to show, e.g. that $\Sigma^7_{1, 2}$ is not diffeomorphic to $S^7$. A little later Kervaire and Milnor {{cite|Kervaire&Milnor1963}} proved that $\Theta_7 \cong \Zz_{28}$ and Eells and Kuiper {{cite|Eells&Kuiper1962}} defined a refinement of the $\lambda$-invariant, now called the Eells-Kuiper $\mu$-invariant, which in particular gives<br />
$$ \Sigma^7_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \in bP_8 \cong \Theta_7.$$<br />
<br />
Shimada {{cite|Shimada1957}} used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres. In this case $\pi_7(SO(8)) \cong \Zz \oplus \Zz$ and the bundle $(m, n)$ has Euler number $n$ and second Pontrjagin class $6(n+2m)$. Moreover $\Theta_{15} \cong \Zz_{8,128} \oplus \Zz_2$ where the $\Zz_{8,128}$-summand is $bP_{16}$ as explained below. Results of {{cite|Wall1962a}} and {{cite|Eells&Kuiper1962}} combine to show that<br />
$$ \Sigma^{15}_{m, 1} = -(m(m-1)/2)\cdot \Sigma_M \cong bP_{16} \subset \Theta_{15}.$$<br />
<br />
*By Adams' solution of the [[Wikipedia:Hopf_invariant| Hopf-invariant]] 1 problem, {{cite|Adams1958}} and {{cite|Adams1960}}, the dimensions n = 3, 7 and 15 are the only dimensions in which a topological n-sphere can be fibre over an m-sphere for 0 < m < n. <br />
</wikitex><br />
<br />
=== Twisting ===<br />
<wikitex>;<br />
By {{cite|Cerf1970}} and {{cite|Smale1962a}} there is an isomorphism $\Theta_{n+1} \cong \Gamma_{n+1}$ for $n \geq 5$ where $\Gamma_{n+1} = \pi_0(\Diff_+(S^n))$ is the group of isotopy classes of orientation preserving diffeomorphisms of $S^n$. The map is given by<br />
$$ \Gamma_{n+1} \to \Theta_{n+1}, ~~~~~[f] \longmapsto \Sigma_{f} := D^{n+1} \cup_f (-D^{n+1}).$$<br />
Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of $S^n$ which are not isotopic to the identity. We give such a construction which probably goes back to Milnor: so far the earliest reference found is the problem list of the 1963 Seattle topology conference {{cite|Lashof1965}}.<br />
<br />
Represent $\alpha \in \pi_p(SO(q))$ and $\beta \in \pi_q(SO(p))$ by smooth compactly supported functions $\alpha : \Rr^p \to SO(q)$ and $\beta : \Rr^q \to SO(p)$ and define the following self-diffeomorphisms of $\Rr^p \times \Rr^q$<br />
$$ F_\alpha : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (x, \alpha(x)y),$$<br />
$$ F_\beta : \Rr^p \times \Rr^q \cong \Rr^p \times \Rr^q, ~~~~(x, y) \longmapsto (\beta(y)x,y),$$<br />
$$s(\alpha, \beta) := F_\alpha F_\beta F_\alpha^{-1}F_\beta^{-1}.$$<br />
If follows that $s(\alpha, \beta)$ is compactly supported and so extends uniquely to a diffeomorphism of $S^{p+q}$. In this way we obtain a bilinear pairing<br />
$$ \sigma : \pi_q(SO(q)) \otimes \pi_q(SO(p)) \longrightarrow \Theta_{p+q+1}, ~~~~(\alpha, \beta) \longmapsto \Sigma_{s(\alpha, \beta)}$$<br />
such that<br />
$$ \sigma(\alpha, \beta) \equiv \Sigma(S\alpha, S\beta).$$<br />
In particular for $k=1, 2$ we see that $\sigma(\gamma_{4k-1}', \gamma_{4k-1}') \cong -\Sigma_M$ generates $bP_{4k}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
Finding invariants of exotic sphere $\Sigma$ which distinguish it from the standard sphere is rather a subtle undertaking. Moreover such invariants are often defined via a manifold $W$ with $\partial W \cong \Sigma$. In this case finding an intrinsic definition and or computation of the relevant invariant can also be subtle.<br />
<br />
We begin by listing some invariants which are equal for all exotic spheres.<br />
{{beginthm|Proposition}} <br />
Let $\Sigma$ be a closed smooth manifold homeomorphic to the n-sphere. Then<br />
# there is an isomorphism of tangent bundles $T\Sigma \cong TS^n$,<br />
# the signature of $\Sigma$ vanishes,<br />
# the Kervaire invariant of $(\Sigma, F)$ is zero for every framing of $\Sigma$.<br />
(To make sense of the first statement remember that the topological space underlying every exotic sphere is homeomorphic to $S^n$.)<br />
{{endthm}}<br />
{{beginrem|Remark}}<br />
The analogue of the first statement for the stable tangent bundle was proven in \cite{Kervaire&Milnor1963|Theorem 3.1}. A proof of the unstable statement is given in \cite{Ray&Pedersen1980|Lemma 1.1}. The next two statements are obvious since both the signature and Kervaire invariant are defined to be zero if $n = 2k+1$ and via a symmetric or quadratic form on $H_k(\Sigma; \Zz) = 0 $ if $n = 2k$.<br />
{{endrem}} <br />
</wikitex><br />
=== Bordism classes ===<br />
<wikitex>;<br />
As every homotopy sphere is stably parallelisable, homotopy spheres admit [[B-Bordism|$B$-structures]] for any $B$. If $B$ is such that $[S^n, F] \mapsto 0 \in \Omega_n^B$ for any stable framing $F$ of $S^n$, then we obtain a well-defined homomorphism<br />
$$ \eta^B : \Theta_n \longrightarrow \Omega_n^B, ~~~\Sigma \longmapsto [\Sigma, F].$$ <br />
* If $B = BO\langle k \rangle$ for $[n/2] + 1 < k < n+2$ then $\Omega_n^B$ is isomorphic to almost framed bordism and the homomorphism $\eta^B$ is the same thing as the $\eta: \Theta_n \to \pi_n(G/O)$ in Theorem \ref{thm-ses}.<br />
* Perhaps surprisingly $\eta_n^{\Spin} \neq 0$ for all $n = 8k+1, 8k+2$, as explained in the next subsection.<br />
* In general determining $\eta^B$ is a hard an interesting problem. <br />
* $B$-coboundaries for elements of $Ker(\eta^B_n)$ are often used to define invariants of $B$-null bordant homotopy spheres.<br />
</wikitex><br />
<br />
=== The α-invariant ===<br />
<wikitex>;<br />
In dimensions $n > 1$, every exotic sphere $\Sigma$ has a unique Spin structure and from above we have the homomorphism $\eta_n^{\Spin} : \Theta_n \to \Omega_n^{\Spin}$. Recall the $\alpha$-invariant homomorphism $\alpha : \Omega_*^{\Spin} \to KO^{-*}$ and that there are isomorphisms $KO^{-8k-1} \cong KO^{-8k-2} \cong \Zz/2$ for all $k \geq 1$.<br />
{{beginthm|Theorem|\cite{Anderson&Brown&Peterson1967}}}<br />
We have $\eta_n^{\Spin}(\Sigma) = 0$ if and only if $\alpha \circ \eta_n^{\Spin}(\Sigma) = 0$ and $\eta_n^{\Spin} \neq 0$ if and only if $n = 8k+1$ or $8k+2$. <br />
{{endthm}}<br />
{{beginrem|Remark}}<br />
Exotic spheres $\Sigma$ with $\alpha(\Sigma) \neq 0$ are often called Hitchin spheres, after \cite{Hitchin1974}: see the discussion of curvature [[#Curvature on exotic spheres|below]].<br />
{{endrem}}<br />
</wikitex><br />
<br />
=== The Eels-Kuiper invariant === <br />
<wikitex>;<br />
</wikitex><br />
=== The s-invariant ===<br />
<wikitex>;<br />
</wikitex><br />
<br />
== Classification ==<br />
<wikitex>;<br />
For $n =1, 2$ and $3$, $\Theta_n = \{ S^n \}$. For $n = 4$, $\Theta_4$ is unknown. We therefore concentrate on higher dimensions.<br />
<br />
For $n \geq 5$, the group of exotic n-spheres $\Theta_n$ fits into the following long exact sequence, first discovered in {{cite|Kervaire&Milnor1963}} (more details can also be found in {{cite|Levine1983}} and {{cite|Lück2001}}):<br />
$$ \dots \stackrel{\eta_{n+1}}{\longrightarrow} \pi_{n+1}(G/O) \stackrel{\sigma_{n+1}}{\longrightarrow} L_{n+1}(e) \stackrel{\omega_{n+1}}{\longrightarrow} \Theta_n \stackrel{\eta_n}{\longrightarrow} \pi_n(G/O) \stackrel{\sigma_n}{\longrightarrow} L_n(e) \to \dots~.$$<br />
Here $L_n(e)$ is the n-th [[Wikipedia:L-theory|L-group]] of the the trivial group: $L_n(e) = \Zz, 0, \Zz/2, 0$ as n = 0, 1, 2 or 3 modulo 4 and the sequence ends at $L_5(e) = 0$. Also $O$ is the stable orthogonal group and $G$ is the stable group of homtopy self-equivalences of the sphere. There is a fibration $O \to G \to G/O$ and the groups $\pi_n(G/O)$ fit into the homtopy long exact sequence<br />
$$ \dots \to \pi_n(O) \to \pi_n(G) \to \pi_n(G/O) \to \pi_{n-1}(O) \to \pi_{n-1}(G) \to \dots $$<br />
of this fibration. The homomorphism $J_n: \pi_n(O) \to \pi_n^S \cong \pi_n(G)$ is the [[Wikipedia:J-homomorphism|stable J-homomorphism]]. In particular, by {{cite|Serre1951}} the groups $\pi_i(G)$ are finite and by {{cite|Bott1959}}, {{cite|Adams1966}} and {{cite|Quillen1971}} the domain, image and kernel of $J_n$ have been completely determined. An important result in {{cite|Kervaire&Milnor1963}} is that the homomorphism $\sigma_{4k}$ is nonzero. The above sequence then gives<br />
<br />
{{beginthm|Theorem|{{cite|Kervaire&Milnor1963}}}}\label{thm-ses}<br />
For $n \geq 5$, the group $\Theta_n$ is finite. Moreover there is an exact sequence<br />
$$ 0 \longrightarrow bP_{n+1} \longrightarrow \Theta_{n} \longrightarrow Coker(J_n) \stackrel{K_n}{\longrightarrow} C_n \longrightarrow 0 $$<br />
where $bP_{n+1} := {Im}(\omega_{n+1})$, the group of homotopy spheres bounding paralellisable manifolds, is a finite cyclic group which vanishes if $n$ is even. Moreover $C_n = 0$ unless $n = 4k+2$ when it is $0$ or $\Zz/2$.<br />
{{endthm}}<br />
<br />
The groups $Coker(J_n)$ are known for $n$ up to approximately 62. In general their determination is a very hard problem. Modulo this problem we see two remaining problems in the determination of $\Theta_n$: an extension problem and the comptutation of the order of the groups $bP_{n+1}$ and $C_n$. We discuss these in turn.<br />
<br />
{{beginthm|Theorem|{{cite|Brumfiel1968}}, {{cite|Brumfiel1969}}, {{cite|Brumfiel1970}}}}<br />
If $n \neq 2^{j} - 3$ the Kervaire-Milnor extension splits: <br />
$$\Theta_n \cong bP_{n+1} \oplus Ker(K_n).$$<br />
{{endthm}}<br />
<br />
The map $K_{4k+2} : Coker(J_{4k+2}) \to \Zz/2$ is the Kervaire invariant and by definition $C_{4k+2} = Im(K_{4k+2})$. By the long exact sequence above we have<br />
<br />
{{beginthm|Theorem|{{cite|Kervaire&Milnor1963|Section 8}}}}<br />
The group $bP_{4k+2}$ is either $\Zz/2$ or $0$. Moreover the following are equivalent:<br />
* $bP_{4k+2} = 0$,<br />
* the Kervaire sphere $\Sigma^{4k+1}_K$ is diffeomorphic to the standard sphere,<br />
* there is a framed manifold with Kervaire invariant 1: $C_{4k+2} \cong \Zz/2$.<br />
Conversely the following are equivalent:<br />
* $bP_{4k+2} = \Zz/2$,<br />
* the Kervaire sphere $\Sigma^{4k+1}_K$ is not diffeomorphic to the standard sphere,<br />
* there is no framed manifold with Kervaire invariant 1: $C_{4k+2} \cong 0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
=== The orders of bP<sub>4k</sub> and bP<sub>4k+2</sub> ===<br />
<wikitex>;<br />
The group $bP_{4k}$ is a cyclic group whose order can be determined using the Hirzebruch's signature theorem if one knows the order of $Im(J_{4k-1}) \subset \pi_{4k-1}(G)$. Adams determined the latter group up to a factor of two which was settled by Quillen with a positive solution to the Adams conjecture. <br />
<br />
{{beginthm|Theorem}}<br />
Let $a_k = (3-(-1)^k)/2$, let $B_k$ be the k-th Bernoulli number (topologist indexing) and for $x \in \Qq$ let $Num(x)$ denote the numerator of $x$ expressed in lowest form. Then for $k \geq 2$, the order of $bP_{4k}$ is<br />
$$ t_k = a_k \cdot 2^{2k-2}\cdot (2^{2k-1}-1) \cdot Num(B_k/4k).$$<br />
{{endthm}}<br />
<br />
{{beginrem|Remark}}<br />
Note that $Num(B_k/4k)$ is odd so the 2-primary order of $bP_{4k}$ is $a_k \cdot 2^{2k-2}$ while the odd part is $(2^{2k-1}-1) \cdot Num(B_k/4k)$. Modulo the Adams conjecture the proof appeared in {{cite|Kervaire&Milnor1963|Section 7}}. Detailed treatments can also be found in {{cite|Levine1983|Section 3}} and {{cite|Lück2001|Chapter 6}}.<br />
{{endrem}}<br />
<br />
The next theorem describes the situation for $bP_{4k+2}$ which is now almost completely understood as well. References for the theorem are given in the remark which follows it.<br />
<br />
{{beginthm|Theorem}}<br />
The group $bP_{4k+2}$ is given as follows:<br />
* $bP_{6} = bP_{14} = bP_{30} = bP_{62} = 0$,<br />
* $bP_{126} = 0$ or $\Zz/2$,<br />
* $bP_{4k+2} = \Zz/2$ else.<br />
{{endthm}}<br />
<br />
{{beginrem|Remark}}<br />
The following is a chronological list of determinations of $bP_{4k+2}$:<br />
* $bP_{10} = \Zz/2$, {{cite|Kervaire1960a}}.<br />
* $bP_{6} = bP_{14} = 0$ {{cite|Kervaire&Milnor1963}}.<br />
* $bP_{8k+2} = \Zz/2$, {{cite|Anderson&Brown&Peterson1966a}}.<br />
* $bP_{30} = 0$, {{cite|Mahowald&Tangora1967}}.<br />
* $bP_{4k+2} = \Zz/2$ unless $4k+2 = 2^j - 2$ {{cite|Browder1969}}.<br />
* $bP_{62} = 0$, {{cite|Barratt&Jones&Mahowald1984}}.<br />
* $bP_{2^j - 2} = \Zz/2$ for $j \geq 8$, {{cite|Hill&Hopkins&Ravenel2009}}.<br />
{{endrem}}<br />
</wikitex><br />
<br />
== Further discussion ==<br />
=== Curvature on exotic spheres ===<br />
<wikitex>;<br />
Gromoll-Meyer proved that a certain exotic 7-sphere can be realized as a biquotient of the compact Lie group Sp(2) and has a Riemannian metric of nonpositive sectional curvature. It is not known whether there exist exotic spheres with Riemannian metrics of positive sectional curvature. For a recent review of which exotic spheres admit metrics of various sorts of positive curvature see \cite{Joachim&Wraith2008}.<br />
</wikitex><br />
<br />
=== The Kervaire-Milnor braid ===<br />
<wikitex>;<br />
$$<br />
\def\curv{1.5pc}% Adjust the curvature of the curved arrows here<br />
\xymatrix@!R@!C@!0@R=2.5pc@C=4pc{% Adjust the spacing here<br />
\pi_n(\Top/O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(O) \ar[dr] \ar@/u\curv/[rr] && \pi_{n-1}(G) \\<br />
& \pi_n(G/O) \ar[dr] \ar[ur] && \pi_{n-1}(\Top) \ar[dr] \ar[ur] \\<br />
\pi_n(G) \ar[ur] \ar@/d\curv/[rr] && \pi_n(G/\Top) \ar[ur] \ar@/d\curv/[rr] && \pi_{n-1}(\Top/O)<br />
}<br />
$$<br />
</wikitex><br />
== PL manifolds admitting no smooth structure ==<br />
<wikitex>;<br />
Let $W^{2n}$ be a [[Exotic spheres#Plumbing|plumbing manifold]] as described above. By a simple version of the [[Alexander trick]], there is a homemorphism $f \colon \partial W \cong S^{2n-1}$ and so we can form the closed topological manifold<br />
$$ \bar W : = W \cup_f D^{2n}.$$<br />
If $\partial W$ is exotic then it turns out that $\bar W$ is a topological manifold which admits no smooth structure! <br />
<br />
\cite{Kervaire1960a} shows that $\bar W^{10}$ is non-smoothable and the arugments there work for all odd $n$ so long as the Kervaire sphere is exotic. <br />
<br />
When $n$ is even the proof is more complicated: one first need's Novikov's theorem that the rational Pontrjagin classes of a topological manifold are homeomorphism invariants \cite{Novikov1965b}. Prior to Novikvo's result, some weaker statements were known. For example, when $n=4$ and $W$ is the total space of a [[Exotic spheres#Sphere bundles|$D^4$-bundle]] over $S^4$ as above and if $\partial W = \Sigma_{m, 1}$ then by \cite{Tamura1961} $\bar W$ is smoothable if and only if $m(m-1)/2 \equiv 0$ mod $4$.</wikitex><ref>Note that Tamura uses a different identification <tex>\pi_3(SO(4)) \cong \Zz \oplus \Zz</tex> from the one used above.</ref><wikitex> Applying Novikov's theorem we know that $\bar W$ is smoothable if and only if $m(m-1)/2 \equiv 0$ mod $56$.<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
== Footnotes ==<br />
<references/><br />
<br />
== External links ==<br />
* The Wikipedia page on [[Wikipedia:Exotic sphere|exotic spheres]]<br />
* The tabulation of the order of the group of exotic spheres in the [http://oeis.org/classic/A001676 On-Line Encyclopedia of Integer Sequences]<br />
* Andrew Ranicki's exotic sphere home page, with many of the original papers: [http://www.maths.ed.ac.uk/~aar/exotic.htm http://www.maths.ed.ac.uk/~aar/exotic.htm] <br />
** Including some [http://www.maths.ed.ac.uk/~aar/papers/km-it.pdf original correspondence between Kervaire and Milnor]<br />
*[http://www.nilesjohnson.net/seven-manifolds.html An animation of exotic 7-spheres]. Slides from a presentation by [http://www.nilesjohnson.net/ Nile Johsnon] at the [http://www.ima.umn.edu/2011-2012/SW1.30-2.1.12/ Second Abel conference] in honor of [[Wikipedia:John Milnor|John Milnor]].<br />
[[Category:Manifolds]]<br />
[[Category:Highly-connected manifolds]]<br />
[[Category:Surgery]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Geometric_3-manifoldsGeometric 3-manifolds2012-03-23T01:50:16Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Let a group $G$ act on a manifold $X$ by homeomorphisms. <br />
<br />
A $\left(G,X\right)$-manifold is a manifold $M$ with a $\left(G,X\right)$-atlas, that is, a collection $\left\{\left(U_i,\phi_i\right):i\in I\right\}$ of homeomorphisms <br />
$$\phi_i:U_i\rightarrow \phi_i\left(U_i\right)\subset X$$ <br />
onto open subsets of $X$ such that all coordinate changes <br />
$$\gamma_{ij}=\phi_i\phi_j^{-1}:\phi_i\left(U_i\cap U_j\right)\rightarrow \phi_j\left(U_i\cap U_j\right)$$<br />
are restrictions of elements of $G$.<br />
<br />
Fix a basepoint $x_0\in M$ and a chart $\left(U_0,\phi_0\right)$ with $x_0\in U_0$. Let $\pi:\widetilde{M}\rightarrow M$ be the universal covering. These data determine the developing map $$D:\widetilde{M}\rightarrow X$$ that agrees with the analytic continuation of $\phi_0\pi$ along each path, in a neighborhood of the path's endpoint.<br />
<br />
If we change the initial data $x_0$ and $\left(U_0,\phi_0\right)$, the developing map $D$ changes by composition with an element of $G$.<br />
<br />
If $\sigma\in\pi_1\left(M,x_0\right)$, analytic continuation along a loop representing $\sigma$ gives a chart $\phi_0^\sigma$ that is comparable to $\phi_0$, since they are both defined at $x_0$. Let $g_\sigma$ be the element of $G$ such that $\phi_0^\sigma=g_\sigma\phi_0$. The map $$H:\pi_1\left(M,x_0\right)\rightarrow G, H\left(\sigma\right)=g_\sigma$$<br />
is a group homomorphism and is called the holonomy of $M$.<br />
<br />
If we change the initial data $x_0$ and $\left(U_0,\phi_0\right)$, the holonomy homomorphisms $H$ changes by conjugation with an element of $G$.<br />
<br />
A $\left(G,X\right)$-manifold is complete if the developing map $D:\widetilde{M}\rightarrow X$ is surjective.<br />
<br />
{{cite|Thurston1997}} Section 3.4<br />
<br />
{{beginthm|Definition}} <br />
A model geometry $\left(G,X\right)$ is a smooth manifold $X$ together with a Lie group of diffeomorphisms of $X$, such that:<br />
<br />
a) $X$ is connected and simply connected;<br />
<br />
b) $G$ acts transitively on $X$, with compact point stabilizers;<br />
<br />
c) $G$ is not contained in any larger group of diffeomorphisms of $X$ with compact point stabilizers;<br />
<br />
d) there exists at least one compact $\left(G,X\right)$-manifold.<br />
{{endthm}}<br />
{{cite|Thurston1997}} Definition 3.8.1<br />
<br />
A 3-manifold is said to be a geometric manifold if it is a $\left(G,X\right)$-manifold for a 3-dimensional model geometry $\left(G,X\right)$.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
{{beginthm|Theorem}}There are eight 3-dimensional model geometries:<br />
<br />
- the round sphere: $X=S^3, G=O(4)$<br />
<br />
- Euclidean space: $X={\mathbb R}^3, G={\mathbb R}^3\rtimes O(3)$<br />
<br />
- hyperbolic space: $X= H^3, G=PSL\left(2,{\mathbb C}\right)\rtimes {\mathbb Z}/2{\mathbb Z}$<br />
<br />
- $X=S^2\times {\mathbb R}, G=O(3)\times \left({\mathbb R}\rtimes {\mathbb Z}/2{\mathbb Z}\right)$<br />
<br />
- $X={\mathbb H}^2\times {\mathbb R}, G=\left(PSL\left(2,{\mathbb R}\right)\rtimes{\mathbb Z}/2{\mathbb Z}\right)\times \left({\mathbb R}\rtimes {\mathbb Z}/2{\mathbb Z}\right)$<br />
<br />
- the universal covering of the unit tangent bundle of the hyperbolic plane: $G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$<br />
<br />
- the Heisenberg group: $G=X=Nil=\left\{\left(\begin{matrix}1&x&z\\0&1&y\\0&0&1\end{matrix}\right):x,y,z\in{\mathbb R}\right\}$<br />
<br />
- the 3-dimensional solvable Lie group $G=X=Sol={\mathbb R}^2\rtimes {\mathbb R}$ with conjugation $t\rightarrow\left(\begin{matrix}e^t&0\\0&e^{-t}\end{matrix}\right)$.{{endthm}}<br />
{{cite|Thurston1997}} Section 3.8<br />
<br />
Outline of Proof:<br />
<br />
Let $G^\prime$ be the connected component of the identity of $G$, and let $G_x^\prime$ be the stabiliser of $x\in X$. <br />
$G^\prime$ acts transitively and $G_x^\prime$ is a closed, connected subgroup of $SO\left(3\right)$.<br />
<br />
Case 1: $G_x^\prime=SO\left(3\right)$. Then $X$ has constant sectional curvature. The Cartan Theorem implies that (up to rescaling) $X$ is isometric to one of $S^3, {\mathbb R}^3, H^3$.<br />
<br />
Case 2: $G_x^\prime \simeq SO\left(2\right)$. Let $V$ be the $G^\prime$-invariant vector field such that, for each $x\in X$, the direction of $V_x$ is the rotation axis of $G_x^\prime$. $V$ descends to a vector field on compact $\left(G,X\right)$-manifolds, therefore the flow of $V$ must preserve volume. In our setting this implies that the flow of $V$ acts by isometries. Hence the flowlines define a 1-dimensional foliation ${\mathcal{F}}$ with embedded leaves. The quotient $X/{\mathcal{F}}$ is a 2-dimensional manifold, which inherits a Riemannian metric such that $G^\prime$ acts transitively by isometries. Thus $Y:=X/{\mathcal{F}}$ has constant curvature and is (up to rescaling) isometric to one of $S^2, {\mathbb R}^2, H^2$. $X$ is a pricipal bundle over $Y$ with fiber ${\mathbb R}$ or $S^1$,<br />
The plane field $\tau$ orthogonal to $\mathcal{F}$ has constant curvature, hence it is either a foliation or a contact structure.<br />
<br />
Case 2a: $\tau$ is a foliation. Thus $X$ is a flat bundle over $Y$. $Y$ is one of $S^2, {\mathbb R}^2, H^2$, hence $\pi_1Y=0$, which implies that $X=Y\times {\mathbb R}$.<br />
<br />
Case 2b: $\tau$ is a contact structure. <br />
For $Y=S^2$ one would obtain for $G$ the group of isometries of $S^3$ that preserve the Hopf fibration. This is not a maximal group with compact stabilizers, thus there is no model geometry in this case.<br />
For $Y={\mathbb R}^2$ one obtains $G=X=Nil$. Namely, $G$ is the subgroup of the group of automorphisms of the standard contact structure $dz-xdy=0$ on ${\mathbb R}^3$ consisting of those automorphisms which are lifts of isometries of the x-y-plane.<br />
For $Y={\mathbb H}^2$ one obtains $G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$.<br />
<br />
Case 3: $G_x^\prime=1$. Then $X=G^\prime/G_x^\prime=G^\prime$ is a Lie group. The only 3-dimensional unimodular Lie group which is not subsumed by one of the previous geometries is $G=Sol$.<br />
<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<br />
A closed 3-manifold is called:<br />
<br />
- irreducible, if every embedded 2-sphere bounds an embedded 3-ball,<br />
<br />
- geometrically atoroidal, if there is no embedded incompressible torus,<br />
<br />
- homotopically atoroidal, if there is no immersed incompressible torus.<br />
<br />
<wikitex>;<br />
{{beginthm|Theorem|(Geometrization)}}<br />
<br />
Let $M$ be a closed, orientable, irreducible, geometrically atoroidal 3-manifold.<br />
<br />
a) If $M$ is homotopically atoroidal, then it admits an $H^3$-geometry.<br />
<br />
b) If $M$ is not homotopically atoroidal, then it admits (at least) one of the seven non-$H^3$-geometries.<br />
{{endthm}}<br />
<br />
{{beginthm|Example|(Geometrization of mapping tori)}} <br />
<br />
Let $\Phi:\Sigma_g\rightarrow \Sigma_g$ be an orientation-preserving homeomorphism of the surface of genus $g$.<br />
<br />
a) If $g=1$, then the mapping torus $M_\Phi$ satisfies the following:<br />
<br />
1. If $\Phi$ is periodic, then $M_\Phi$ admits an ${\mathbb R}^3$ geometry.<br />
<br />
2. If $\Phi$ is reducible, then $M_\Phi$ contains an embedded incompressible torus.<br />
<br />
3. If $\Phi$ is Anosov, then $M_\Phi$ admits a $Sol$ geometry.<br />
<br />
b) If $g\ge 2$, then the mapping torus $M_\Phi$ satisfies the following:<br />
<br />
1. If $\Phi$ is periodic, then $M_\Phi$ admits an $H^2\times{\mathbb R}$-geometry.<br />
<br />
2. If $\Phi$ is reducible, then $M_\Phi$ contains an embedded incompressible torus.<br />
<br />
3. If $\Phi$ is pseudo-Anosov, then $M_\Phi$ admits an $H^3$-geometry.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hyperbolic_3-manifoldsHyperbolic 3-manifolds2011-10-29T09:10:56Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
A 3-manifold is hyperbolic if it satisfies the following (equivalent) conditions: <br />
<br>- it admits a complete Riemannian metric of sectional curvature constant -1,<br />
<br><br />
- it admits a Riemannian metric such that its universal covering (with the pull-back metric) is isometric to hyperbolic 3-space ${\Bbb H}^3$,<br />
<br><br />
- it is homeomorphic to $\Gamma\backslash{\Bbb H}^3$, for some discrete, torsion-free group of isometries of hyperbolic 3-space.<br />
<br><br />
<br>Here, hyperbolic 3-space ${\Bbb H}^3$ is the simply connected, complete, Riemannian manifold with sectional curvature constant -1. By Cartan's Theorem, ${\Bbb H}^3$ is unique up to isometry. There are different models for ${\Bbb H}^3$, like the upper half-space model, the Poincaré disc model or the hyperboloid model. <br />
<br>The ideal boundary $\partial_\infty{\Bbb H}^3$ can be identified with the projective line $P^1{\Bbb C}\cong S^2$. Isometries of hyperbolic 3-space act as conformal automorphisms of the ideal boundary. Thus we can identify the isometry group $Isom\left({\Bbb H}^3\right)$ with the group of conformal automorphisms $Conf\left(S^2\right)$.<br />
<br><br />
<br>The group $SO\left(3,1\right)$ acts on the hyperboloid model and one can use this action to identify $SO\left(3,1\right)$ with the index two subgroup $Isom^+\left({\Bbb H}^3\right)\subset Isom\left({\Bbb H}^3\right)$ of orientation-preserving isometries. The action is transitive and has $SO\left(3\right)$ as a point stabilizer, thus ${\Bbb H}^3$ is isometric to the homogeneous space $SO\left(3,1\right)/SO\left(3\right)\cong Spin\left(3,1\right)/Spin\left(3\right)$.<br />
<br>The group $PSL\left(2,{\Bbb C}\right)$ acts by fractional-linear automorphisms on $P^1{\Bbb C}$. This action on $\partial_\infty{\Bbb H}^3$ uniquely extends to an action on ${\Bbb H}^3$ by orientation-preserving isometries. One can use this action to identify $PSL\left(2,{\Bbb C}\right)$ with <br />
$Conf^+\left(S^2\right)=Isom^+\left({\Bbb H}^3\right)$.<br />
The action is transitive and has $PSU\left(2\right)$ as a point stabilizer, thus ${\Bbb H}^3$ is isometric to the homogeneous space $PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)\cong SL\left(2,{\Bbb C}\right)/SU\left(2\right)$.<br />
<br><br />
<br>Thus, if $M$ is oriented, then there are two more equivalent conditions:<br />
<br>An oriented 3-manifold is hyperbolic if and only if <br />
<br>- it is homeomorphic to $\Gamma\backslash PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)$ for some discrete, torsion-free subgroup $\Gamma\subset PSL\left(2,{\Bbb C}\right)$,<br />
<br>- <br />
it is homeomorphic to $\Gamma\backslash SO\left(3,1\right)/SO\left(3\right)$ <br />
for some discrete, torsion-free subgroup $\Gamma\subset SO\left(3,1\right)$.<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<br />
<wikitex>;<br />
...<br />
</wikitex><br />
== Invariants ==<br />
<wikitex>;<br />
By Mostow rigidity, complete hyperbolic metrics of finite volume on a 3-manifold are unique up to isometry. This implies that geometric invariants of the hyperbolic metric, such as the volume and the Chern-Simons-invariant, are topological invariants. <br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<wikitex>;<br />
By the Marden tameness conjecture (proved by Agol and Calegari-Gabai) each hyperbolic 3-manifold with finitely generated fundamental group is the interior of a compact 3-manifold with boundary.<br />
<br><br />
<br>If $M$ is an orientable 3-manifold with boundary, whose interior admits a complete hyperbolic metric of finite volume, then $\partial M$ is a (possibly empty) union of incompressible tori.<br />
<br><br />
<br>Ends of infinite volume fall into two classes, geometrically finite ends and geometrically infinite ends....<br />
<br />
Geometrically finite ends are classified by the corresponding points in Teichmüller space of $\partial M$. (Ahlfors-Bers) ...<br />
<br />
Geometrically infinite ends are classified by the corresponding ending laminations. (Brock-Canary-Minsky) ....<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<!-- == Acknowledgments ==<br />
...<br />
<br />
== Footnotes ==<br />
<references/> --><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- == External links ==<br />
* The Wikipedia page about [[Wikipedia:Page_name|link text]]. --><br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hyperbolic_SurfacesHyperbolic Surfaces2011-10-29T09:10:31Z<p>Kuessner: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
<br />
<div class="thumb tright"><div class="thumbinner" style="width:214px;"><a href="/File:Polygon_construction.png" class="image"><img alt="" src="/images/5/5e/Polygon_construction.png" width="212" height="207" class="thumbimage" /></a><div class="thumbcaption"><br />
<br />
<div class="magnify"><br />
<a href="/File:Polygon_construction.png" class="internal" title="Enlarge"><br />
<img src="/skins/common/images/magnify-clip.png" width="15" height="11" alt="" /><br />
</a><br />
</div>The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.<br />
</div><br />
</div><br />
</div><br />
<br />
Any hyperbolic metric on a closed, orientable surface $S_g$ of genus $g\ge 2$ is obtained by the following <br />
construction: choose a geodesic $4g$-gon in the hyperbolic plane ${\Bbb H}^2$ with area $4(g-1)\pi$. (This implies that the sum of interior angles is $2\pi$.) Then choose orientation-preserving isometries $I_1,J_1,\ldots,I_g,J_g$ which realise the gluing pattern of $S_g$: for $j=1,\ldots,g$ we require that $I_j$ maps $a_j$ to $\overline{a}_j$, $J_j$ maps $b_j$ to $\overline{b}_j$. Let $\Gamma\subset Isom^+\left({\Bbb H}^2\right)$ be the subgroup generated by $I_1,J_1,\ldots,I_g,J_g$. Then $\Gamma$ is a discrete subgroup of $Isom^+\left({\Bbb H}^2\right)$ and $\Gamma\backslash{\Bbb H}^2$ is a hyperbolic surface diffeomorphic to ${\Bbb H}^2$.<br />
<br />
The moduli space of hyperbolic metrics on the closed, orientable surface $S_g$ is $\left(6g-6\right)$-dimensional.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<!-- == Acknowledgments ==<br />
...<br />
<br />
== Footnotes ==<br />
<references/> --><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- == External links ==<br />
* The Wikipedia page about [[Wikipedia:Page_name|link text]]. --><br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hyperbolic_SurfacesHyperbolic Surfaces2011-10-29T08:48:45Z<p>Kuessner: Created page with "<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - Fo..."</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
<br />
Any hyperbolic metric on a closed, orientable surface $S_g$ of genus $g\ge 2$ is obtained by the following <br />
construction: choose a geodesic $4g$-gon in the hyperbolic plane ${\Bbb H}^2$ with area $4(g-1)\pi$. (This implies that the sum of interior angles is $2\pi$.) Then choose orientation-preserving isometries $I_1,J_1,\ldots,I_g,J_g$ which realise the gluing pattern of $S_g$: for $j=1,\ldots,g$ we require that $I_j$ maps $a_j$ to $\overline{a}_j$, $J_j$ maps $b_j$ to $\overline{b}_j$. Let $\Gamma\subset Isom^+\left({\Bbb H}^2\right)$ be the subgroup generated by $I_1,J_1,\ldots,I_g,J_g$. Then $\Gamma$ is a discrete subgroup of $Isom^+\left({\Bbb H}^2\right)$ and $\Gamma\backslash{\Bbb H}^2$ is a hyperbolic surface diffeomorphic to ${\Bbb H}^2$.<br />
<br />
The moduli space of hyperbolic metrics on the closed, orientable surface $S_g$ is $\left(6g-6\right)$-dimensional.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<!-- == Acknowledgments ==<br />
...<br />
<br />
== Footnotes ==<br />
<references/> --><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- == External links ==<br />
* The Wikipedia page about [[Wikipedia:Page_name|link text]]. --><br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hyperbolic_3-manifoldsHyperbolic 3-manifolds2011-08-15T10:04:23Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
A 3-manifold is hyperbolic if it satisfies the following (equivalent) conditions: <br />
<br>- it admits a complete Riemannian metric of sectional curvature constant -1,<br />
<br><br />
- it admits a Riemannian metric such that its universal covering (with the pull-back metric) is isometric to hyperbolic 3-space ${\Bbb H}^3$,<br />
<br><br />
- it is homeomorphic to $\Gamma\backslash{\Bbb H}^3$, for some discrete, torsion-free group of isometries of hyperbolic 3-space.<br />
<br><br />
<br>Here, hyperbolic 3-space ${\Bbb H}^3$ is the simply connected, complete, Riemannian manifold with sectional curvature constant -1. By Cartans Theorem, ${\Bbb H}^3$ is unique up to isometry. There are different models for ${\Bbb H}^3$, like the upper half-space model, the Poincaré disc model or the hyperboloid model. <br />
<br>The ideal boundary $\partial_\infty{\Bbb H}^3$ can be identified with projective space $P^1{\Bbb C}\cong S^2$. Isometries of hyperbolic 3-space act as conformal automorphisms of the ideal boundary. Thus we can identify the isometry group $Isom\left({\Bbb H}^3\right)$ with the group of conformal automorphisms $Conf\left(S^2\right)$.<br />
<br><br />
<br>The group $SO\left(3,1\right)$ acts on the hyperboloid model and one can use this action to identify $SO\left(3,1\right)$ with the index two subgroup $Isom^+\left({\Bbb H}^3\right)\subset Isom\left({\Bbb H}^3\right)$ of orientation-preserving isometries. The action is transitive and has $SO\left(3\right)$ as a point stabilizer, thus ${\Bbb H}^3$ is isometric to the homogeneous space $SO\left(3,1\right)/SO\left(3\right)\cong Spin\left(3,1\right)/Spin\left(3\right)$.<br />
<br>The group $PSL\left(2,{\Bbb C}\right)$ acts by fractional-linear automorphisms on $P^1{\Bbb C}$. This action on $\partial_\infty{\Bbb H}^3$ uniquely extends to an action on ${\Bbb H}^3$ by orientation-preserving isometries. One can use this action to identify $PSL\left(2,{\Bbb C}\right)$ with <br />
$Conf^+\left(S^2\right)=Isom^+\left({\Bbb H}^3\right)$.<br />
The action is transitive and has $PSU\left(2\right)$ as a point stabilizer, thus ${\Bbb H}^3$ is isometric to the homogeneous space $PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)\cong SL\left(2,{\Bbb C}\right)/SU\left(2\right)$.<br />
<br><br />
<br>Thus, if $M$ is oriented, then there are two more equivalent conditions:<br />
<br>An oriented 3-manifold is hyperbolic if and only if <br />
<br>- it is homeomorphic to $\Gamma\backslash PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)$ for some discrete, torsion-free subgroup $\Gamma\subset PSL\left(2,{\Bbb C}\right)$,<br />
<br>- <br />
it is homeomorphic to $\Gamma\backslash SO\left(3,1\right)/SO\left(3\right)$ <br />
for some discrete, torsion-free subgroup $\Gamma\subset SO\left(3,1\right)$.<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
By Mostow rigidity, complete hyperbolic metrics of finite volume on a 3-manifold are unique up to isometry. This implies that geometric invariants of the hyperbolic metric, such as the volume and the Chern-Simons-invariant, are topological invariants. <br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<wikitex>;<br />
By the Marden tameness conjecture (proved by Agol and Calegari-Gabai) each hyperbolic 3-manifold with finitely generated fundamental group is the interior of a compact 3-manifold with boundary.<br />
<br><br />
<br>If $M$ is an orientable 3-manifold with boundary, whose interior admits a complete hyperbolic metric of finite volume, then $\partial M$ is a (possibly empty) union of incompressible tori.<br />
<br><br />
<br>Ends of infinite volume fall into two classes, geometrically finite ends and geometrically infinite ends....<br />
<br />
Geometrically finite ends are classified by the corresponding points in Teichmüller space of $\partial M$. (Ahlfors-Bers) ...<br />
<br />
Geometrically infinite ends are classified by the corresponding ending laminations. (Brock-Canary-Minsky) ....<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<!-- == Acknowledgments ==<br />
...<br />
<br />
== Footnotes ==<br />
<references/> --><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- == External links ==<br />
* The Wikipedia page about [[Wikipedia:Page_name|link text]]. --><br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hirsch-Smale_theoryHirsch-Smale theory2011-07-28T11:49:43Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
An immersion $f:A\rightarrow N$ is a map of manifolds which is locally an embedding, i.e. such that for<br />
each $a \in A$ there exists an open neighbourhood $U \subseteq A$ with $a \in U$ and $f\vert:U \to N$ an embedding.<br />
A regular homotopy of immersions $f_0,f_1:A \rightarrow N$ is a homotopy $h:f_0 \simeq f_1:A \rightarrow N$ <br />
such that each $h_t:A \rightarrow N$ ($t \in I$) is an immersion.<br />
<br />
Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the [[h-principle]].<br />
</wikitex><br />
<br />
== Results ==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} For a submanifold $A\subset{\mathbb R}^q$ and a manifold $N$, a pair $\left(f,f^\prime\right)$ is called an ${\mathbb R}^q$-immersion if <br />
<br />
- $f:A\rightarrow N$ is an immersion,<br />
<br />
- $f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$ is a linear bundle map, and<br />
<br />
- there exists an open neighborhood $U$ of $A$ in ${\mathbb R}^q$ and an immersion $g:U\rightarrow N$ such that $g\mid_{A}=f$ and $Dg\mid_{A}=f^\prime$. <br />
{{endthm}}<br />
<br />
{{beginthm|Definition}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$-immersion. The obstruction to extending $\left(f,f^\prime\right)$, denoted by $\tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right)$ with $V_{n,q}$ the Stiefel manifold of $q$-frames in ${\mathbb R}^n$, is the homotopy class of $$x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).$${{endthm}}<br />
<br />
<br />
{{beginthm|Theorem |}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$-immersion. <br />
<br />
If $k+1<n$ and $\tau\left(f^\prime\right)=0$, then $\left(f,f^\prime\right)$ can be extended to an ${\mathbb R}^q$-immersion $f:D^{k+1}\rightarrow {\mathbb R}^n$. {{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 3.9.<br />
<br />
<br />
This Theorem does not hold for $n=k+1$. <br />
<br />
If $n=k+1=2$, then conditions for the extendibility of $\left(f,f^\prime\right)$ are given in {{cite|Blank1967}}, more details are worked out in {{cite|Frisch2010}}.<br />
<br />
<br />
==Applications==<br />
<br />
{{beginthm|Theorem |}}<br />
Let $M$ be a smooth manifold of dimension $k<n$. Then the following assertions are equivalent:<br />
<br />
(i) $M$ can be immersed into ${\mathbb R}^n$,<br />
<br />
(ii) there exists a $GL\left(k,{\mathbb R}\right)$-equivariant map $T_k\left(M\right)\rightarrow V_{n,k}$, where $T_k\left(M\right)\rightarrow M$ is the $k$-frame bundle and $V_{n,k}$ is the Stiefel manifold,<br />
<br />
(iii) the bundle associated to $T_k\left(M\right)$ with fiber $V_{n,k}$ has a cross section.<br />
{{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 6.1.<br />
The equivalence between (i) and (ii) is proved by induction over the dimension of subsimplices in a triangulation of $M$ using Theorem 3.9 (which can be adapted from $\left(D^k,S^{k-1}\right)$ to $\left(\Delta^k,\partial \Delta^k\right)$) for the inductive step. The equivalence between (ii) and (iii) is a general fact from the theory of fiber bundles.<br />
{{beginthm|Corollary |}} Parallelizable $k$-manifolds can be immersed into ${\mathbb R}^{k+1}$.{{endthm}}<br />
<br />
{{beginthm|Corollary |}} Exotic $7$-spheres can be immersed into ${\mathbb R}^8$.{{endthm}}<br />
<br />
<br />
<br />
==References==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hirsch-Smale_theoryHirsch-Smale theory2011-07-28T11:49:10Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
An immersion $f:A\rightarrow N$ is a map of manifolds which is locally an embedding, i.e. such that for<br />
each $a \in A$ there exists an open neighbourhood $U \subseteq A$ with $a \in U$ and $f\vert:U \to N$ an embedding.<br />
A regular homotopy of immersions $f_0,f_1:A \rightarrow N$ is a homotopy $h:f_0 \simeq f_1:A \rightarrow N$ <br />
such that each $h_t:A \rightarrow N$ ($t \in I$) is an immersion.<br />
<br />
Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the [[h-principle]].<br />
</wikitex><br />
<br />
== Results ==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} For a submanifold $A\subset{\mathbb R}^q$ and a manifold $N$, a pair $\left(f,f^\prime\right)$ is called an ${\mathbb R}^q$-immersion if <br />
<br />
- $f:A\rightarrow N$ is an immersion,<br />
<br />
- $f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$ is a linear bundle map, and<br />
<br />
- there exists an open neighborhood $U$ of $A$ in ${\mathbb R}^q$ and an immersion $g:U\rightarrow N$ such that $g\mid_{A}=f$ and $Dg\mid_{A}=f^\prime$. <br />
{{endthm}}<br />
<br />
{{beginthm|Definition}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$-immersion. The obstruction to extending $\tau\left(f,f^\prime\right)$, denoted by $\tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right)$ with $V_{n,q}$ the Stiefel manifold of $q$-frames in ${\mathbb R}^n$, is the homotopy class of $$x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).$${{endthm}}<br />
<br />
<br />
{{beginthm|Theorem |}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$-immersion. <br />
<br />
If $k+1<n$ and $\tau\left(f^\prime\right)=0$, then $\left(f,f^\prime\right)$ can be extended to an ${\mathbb R}^q$-immersion $f:D^{k+1}\rightarrow {\mathbb R}^n$. {{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 3.9.<br />
<br />
<br />
This Theorem does not hold for $n=k+1$. <br />
<br />
If $n=k+1=2$, then conditions for the extendibility of $\left(f,f^\prime\right)$ are given in {{cite|Blank1967}}, more details are worked out in {{cite|Frisch2010}}.<br />
<br />
<br />
==Applications==<br />
<br />
{{beginthm|Theorem |}}<br />
Let $M$ be a smooth manifold of dimension $k<n$. Then the following assertions are equivalent:<br />
<br />
(i) $M$ can be immersed into ${\mathbb R}^n$,<br />
<br />
(ii) there exists a $GL\left(k,{\mathbb R}\right)$-equivariant map $T_k\left(M\right)\rightarrow V_{n,k}$, where $T_k\left(M\right)\rightarrow M$ is the $k$-frame bundle and $V_{n,k}$ is the Stiefel manifold,<br />
<br />
(iii) the bundle associated to $T_k\left(M\right)$ with fiber $V_{n,k}$ has a cross section.<br />
{{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 6.1.<br />
The equivalence between (i) and (ii) is proved by induction over the dimension of subsimplices in a triangulation of $M$ using Theorem 3.9 (which can be adapted from $\left(D^k,S^{k-1}\right)$ to $\left(\Delta^k,\partial \Delta^k\right)$) for the inductive step. The equivalence between (ii) and (iii) is a general fact from the theory of fiber bundles.<br />
{{beginthm|Corollary |}} Parallelizable $k$-manifolds can be immersed into ${\mathbb R}^{k+1}$.{{endthm}}<br />
<br />
{{beginthm|Corollary |}} Exotic $7$-spheres can be immersed into ${\mathbb R}^8$.{{endthm}}<br />
<br />
<br />
<br />
==References==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Framed_bordismFramed bordism2011-07-08T09:40:30Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<br />
<wikitex>;<br />
The framed bordism groups $\Omega_n^{fr}$ of manifolds with a framing of the stable normal bundle (or equivalently the stable tangent bundle) are isomorphic to the [[wikipedia:Homotopy_groups_of_spheres#Stable_and_unstable_groups|stable homotopy groups of spheres]] $\pi_n^{s}$. <br />
These groups are now completely known only in a range up to 62: They seem to be very complicated, and no general description is known. <br />
(As an illustration: there is $p$-torsion in $\Omega_*^{fr}$ for all primes $p$.)<br />
</wikitex><br />
<br />
== Generators ==<br />
<wikitex>;<br />
* $\Omega_0^{fr}=\Zz$, generated by a point. <br />
<br />
* $\Omega_1^{fr}=\Zz_2$, generated by $S^1$ with the Lie group framing.<br />
<br />
* $\Omega_2^{fr}=\Zz_2$, <br />
<br />
* $\Omega_3^{fr}=\Zz_{24}$, generated by $S^3=SU(2)$ with the Lie group framing of <br />
<br />
* $\Omega_4^{fr}=\Omega_5^{fr}=0$.<br />
<br />
* $\Omega_6^{fr} = \Zz_2$, generated $S^3 \times S^3$ with the Lie group framing.<br />
<br />
* $\Omega_7^{fr} \cong \Zz_{240} \cong \Zz_{16} \oplus \Zz_3 \oplus \Zz_5$, generated by $S^7$ with twisted framing defined by the generator of $\pi_7(O) \cong \Zz$.<br />
<br />
See also:<br />
*[[Wikipedia:Homotopy_groups_of_spheres#Table_of_stable_homotopy_groups|this table]] from the Wikipedia article on homotopy groups of spheres for more values.<br />
*[http://www.math.cornell.edu/~hatcher/stemfigs/stems.html this table] from Allen Hatchers home page.<br />
<br />
Serre {{cite|Serre1951}} proved that $\Omega_n^{fr}$ is a finite abelian group for $n>0$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
Degree of a map $S^n\to S^n$. Since stably framed manifolds have stably trivial tangent bundles, all other characteristic numbers are zero.<br />
</wikitex><br />
<br />
== Classification ==<br />
<wikitex>;<br />
The case of framed bordism is the original case of the [[B-Bordism#The Pontrjagin-Thom isomorphism|Pontrjagin-Thom isomorphism]], discovered by Pontryagin. The Thom spectrum $MPBO$ corresponding to the path fibration over $BO$ is homotopy equivalent to the sphere spectrum $S$ since the path space is contractible. Thus we get $$\Omega_n^{fr} \cong \pi_n(MPBO) \cong \pi_n^s.$$<br />
<br />
Consequently most of the classification results use homotopy theory.<br />
<br />
Adams spectral sequence and Novikov's generalization {{cite|Ravenel1986}}.<br />
Toda brackets.<br />
Nishida {{cite|Nishida1973}} proved that in the ring $\Omega_*^{fr}$ all elements of positive degree are nilpotent.<br />
</wikitex><br />
<br />
== Further topics ==<br />
<wikitex>;<br />
Kervaire invariant 1, Hopf invariant 1 problems, J-homomorphism, first $p$-torsion in degree.<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
[[Category:Bordism]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hirsch-Smale_theoryHirsch-Smale theory2011-07-07T11:45:09Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
An immersion $f:A\rightarrow N$ is a map of manifolds which is locally an embedding, i.e. such that for<br />
each $a \in A$ there exists an open neighbourhood $U \subseteq A$ with $a \in U$ and $f\vert:U \to N$ an embedding.<br />
A regular homotopy of immersions $f_0,f_1:A \rightarrow N$ is a homotopy $h:f_0 \simeq f_1:A \rightarrow N$ <br />
such that each $h_t:A \rightarrow N$ ($t \in I$) is an immersion.<br />
<br />
Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the [[h-principle]].<br />
</wikitex><br />
<br />
== Results ==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} For a submanifold $A\subset{\mathbb R}^q$ and a manifold $N$, a pair $\left(f,f^\prime\right)$ is called an ${\mathbb R}^q$-immersion if <br />
<br />
- $f:A\rightarrow N$ is an immersion,<br />
<br />
- $f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$ is a linear map, and<br />
<br />
- there exists an open neighborhood $U$ of $A$ in ${\mathbb R}^q$ and an immersion $g:U\rightarrow N$ such that $g\mid_{A}=f$ and $Dg\mid_{A}=f^\prime$. <br />
{{endthm}}<br />
<br />
{{beginthm|Definition}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$-immersion. The obstruction to extending $\tau\left(f,f^\prime\right)$, denoted by $\tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right)$ with $V_{n,q}$ the Stiefel manifold of $q$-frames in ${\mathbb R}^n$, is the homotopy class of $$x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).$${{endthm}}<br />
<br />
<br />
{{beginthm|Theorem |}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$-immersion. <br />
<br />
If $k+1<n$ and $\tau\left(f^\prime\right)=0$, then $\left(f,f^\prime\right)$ can be extended to an ${\mathbb R}^q$-immersion $f:D^{k+1}\rightarrow {\mathbb R}^n$. {{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 3.9.<br />
<br />
<br />
This Theorem does not hold for $n=k+1$. <br />
<br />
If $n=k+1=2$, then conditions for the extendibility of $\left(f,f^\prime\right)$ are given in {{cite|Blank1967}}, more details are worked out in {{cite|Frisch2010}}.<br />
<br />
<br />
==Applications==<br />
<br />
{{beginthm|Theorem |}}<br />
Let $M$ be a smooth manifold of dimension $k<n$. Then the following assertions are equivalent:<br />
<br />
(i) $M$ can be immersed into ${\mathbb R}^n$,<br />
<br />
(ii) there exists a $GL\left(k,{\mathbb R}\right)$-equivariant map $T_k\left(M\right)\rightarrow V_{n,k}$, where $T_k\left(M\right)\rightarrow M$ is the $k$-frame bundle and $V_{n,k}$ is the Stiefel manifold,<br />
<br />
(iii) the bundle associated to $T_k\left(M\right)$ with fiber $V_{n,k}$ has a cross section.<br />
{{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 6.1.<br />
The equivalence between (i) and (ii) is proved by induction over the dimension of subsimplices in a triangulation of $M$ using Theorem 3.9 (which can be adapted from $\left(D^k,S^{k-1}\right)$ to $\left(\Delta^k,\partial \Delta^k\right)$) for the inductive step. The equivalence between (ii) and (iii) is a general fact from the theory of fiber bundles.<br />
{{beginthm|Corollary |}} Parallelizable $k$-manifolds can be immersed into ${\mathbb R}^{k+1}$.{{endthm}}<br />
<br />
{{beginthm|Corollary |}} Exotic $7$-spheres can be immersed into ${\mathbb R}^8$.{{endthm}}<br />
<br />
<br />
<br />
==References==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hirsch-Smale_theoryHirsch-Smale theory2011-07-06T18:19:01Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the [[h-principle]].<br />
</wikitex><br />
<br />
== Results ==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} For a submanifold $A\subset{\mathbb R}^q$ and a manifold $N$, a pair $\left(f,f^\prime\right)$ is called an ${\mathbb R}^q$-immersion if <br />
<br />
- $f:A\rightarrow N$ is an immersion,<br />
<br />
- $f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$ is a linear map, and<br />
<br />
- there exists an open neighborhood $U$ of $A$ in ${\mathbb R}^q$ and an immersion $g:U\rightarrow N$ such that $g\mid_{A}=f$ and $Dg\mid_{A}=f^\prime$. <br />
{{endthm}}<br />
<br />
{{beginthm|Definition}}Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$-immersion. The obstruction to extending $\tau\left(f,f^\prime\right)$, denoted by $\tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right)$ with $V_{n,q}$ the Stiefel manifold of $q$-frames in ${\mathbb R}^n$, is the homotopy class of $$x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).$${{endthm}}<br />
<br />
<br />
{{beginthm|Theorem |}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$-immersion. <br />
<br />
If $k+1<n$ and $\tau\left(f^\prime\right)=0$, then $\left(f,f^\prime\right)$ can be extended to an ${\mathbb R}^q$-immersion $f:D^{k+1}\rightarrow {\mathbb R}^n$. {{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 3.9.<br />
<br />
<br />
This Theorem does not hold for $n=k+1$. <br />
<br />
If $n=k+1=2$, then conditions for the extendibility of $\left(f,f^\prime\right)$ are given in {{cite|Blank1967}}, more details are worked out in {{cite|Frisch2010}}.<br />
<br />
<br />
==Applications==<br />
<br />
{{beginthm|Theorem |}}<br />
Let $M$ be a smooth manifold of dimension $k<n$. Then the following assertions are equivalent:<br />
<br />
(i) $M$ can be immersed into ${\mathbb R}^n$,<br />
<br />
(ii) there exists a $GL\left(k,{\mathbb R}\right)$-equivariant map $T_k\left(M\right)\rightarrow V_{n,k}$, where $T_k\left(M\right)\rightarrow M$ is the $k$-frame bundle and $V_{n,k}$ is the Stiefel manifold,<br />
<br />
(iii) the bundle associated to $T_k\left(M\right)$ with fiber $V_{n,k}$ has a cross section.<br />
{{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 6.1.<br />
The equivalence between (i) and (ii) is proved by induction over the dimension of subsimplices in a triangulation of $M$ using Theorem 3.9 (which can be adapted from $\left(D^k,S^{k-1}\right)$ to $\left(\Delta^k,\partial \Delta^k\right)$) for the inductive step. The equivalence between (ii) and (iii) is a general fact from the theory of fiber bundles.<br />
{{beginthm|Corollary |}} Parallelizable $k$-manifolds can be immersed into ${\mathbb R}^{k+1}$.{{endthm}}<br />
<br />
{{beginthm|Corollary |}} Exotic $7$-spheres can be immersed into ${\mathbb R}^8$.{{endthm}}<br />
<br />
<br />
<br />
==References==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hirsch-Smale_theoryHirsch-Smale theory2011-07-06T09:26:10Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the [[h-principle]].<br />
</wikitex><br />
<br />
== Results ==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} For a submanifold $A\subset{\mathbb R}^q$ and a manifold $N$, a pair $\left(f,f^\prime\right)$ is called an ${\mathbb R}^q$-immersion if <br />
<br />
- $f:A\rightarrow N$ is an immersion,<br />
<br />
- $f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$ is a linear map, and<br />
<br />
- there exists an open neighborhood $U$ of $A$ in ${\mathbb R}^q$ and an immersion $g:U\rightarrow N$ such that $g\mid_{A}=f$ and $Dg\mid_{A}=f^\prime$. <br />
{{endthm}}<br />
<br />
{{beginthm|Definition}}Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$-immersion. The obstruction to extending $\tau\left(f,f^\prime\right)$, denoted by $\tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right)$ with $V_{n,q}$ the Stiefel manifold of $q$-frames in ${\mathbb R}^n$, is the homotopy class of $$x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).$${{endthm}}<br />
<br />
<br />
{{beginthm|Theorem |}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$-immersion. <br />
<br />
If $k+1<n$ and $\tau\left(f^\prime\right)=0$, then $\left(f,f^\prime\right)$ can be extended to an ${\mathbb R}^q$-immersion $f:D^{k+1}\rightarrow {\mathbb R}^n$. {{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 3.9.<br />
<br />
<br />
This Theorem does not hold for $n=k+1$. <br />
<br />
If $n=k+1=2$, then conditions for the extendibility of $\left(f,f^\prime\right)$ are given in {{cite|Blank1967}}, more details are worked out in {{cite|Frisch2010}}.<br />
<br />
==References==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Template:Blank1967Template:Blank19672011-07-06T09:20:48Z<p>Kuessner: Created page with "<!-- Please type your bibitem below following: Author(s), ''Title'', Journal/Publisher, date, pages. Here is a sample to edit M. A. Kervaire and J. W. Milnor, ''[URL Groups of ..."</p>
<hr />
<div><!-- Please type your bibitem below following: <br />
Author(s), ''Title'', Journal/Publisher, date, pages. <br />
Here is a sample to edit<br />
M. A. Kervaire and J. W. Milnor, ''[URL Groups of homotopy spheres I]'', Ann. of Math. (2) '''77''' (1963) 504–537. {{Mathscinet|0148075|26 #5584}} {{Zentralblatt|0107.40303}}<br />
Be careful to remove any line-breaks.<br />
--><br />
<wikitex> Samuel Joel Blank, "Extending Immersions and regular Homotopies in Codimension 1", PhD Thesis Brandeis University, 1967<br />
<br />
{{Mathscinet| | }} {{Zentralblatt| }}<br />
</wikitex><br />
<noinclude>[[Category:Bibliography]]</noinclude></div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Template:Frisch2010Template:Frisch20102011-07-06T09:18:47Z<p>Kuessner: Created page with "<!-- Please type your bibitem below following: Author(s), ''Title'', Journal/Publisher, date, pages. Here is a sample to edit M. A. Kervaire and J. W. Milnor, ''[URL Groups of ..."</p>
<hr />
<div><!-- Please type your bibitem below following: <br />
Author(s), ''Title'', Journal/Publisher, date, pages. <br />
Here is a sample to edit<br />
M. A. Kervaire and J. W. Milnor, ''[URL Groups of homotopy spheres I]'', Ann. of Math. (2) '''77''' (1963) 504–537. {{Mathscinet|0148075|26 #5584}} {{Zentralblatt|0107.40303}}<br />
Be careful to remove any line-breaks.<br />
--><br />
<wikitex> Dennis Frisch, "Classification of Immersions which are bounded by Curves in Surfaces", PhD Thesis TU Darmstadt, 2010<br />
<br />
{{Mathscinet| | }} {{Zentralblatt| }}<br />
</wikitex><br />
<noinclude>[[Category:Bibliography]]</noinclude></div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hirsch-Smale_theoryHirsch-Smale theory2011-07-05T14:24:46Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the [[h-principle]].<br />
</wikitex><br />
<br />
== Results ==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} For a submanifold $A\subset{\mathbb R}^q$ and a manifold $N$, a pair $\left(f,f^\prime\right)$ is called an ${\mathbb R}^q$-immersion if <br />
<br />
- $f:A\rightarrow N$ is an immersion,<br />
<br />
- $f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$ is a linear map, and<br />
<br />
- there exists an open neighborhood $U$ of $A$ in ${\mathbb R}^q$ and an immersion $g:U\rightarrow N$ such that $g\mid_{A}=f$ and $Dg\mid_{A}=f^\prime$. <br />
{{endthm}}<br />
<br />
{{beginthm|Definition}}Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$-immersion. The obstruction to extending $\tau\left(f,f^\prime\right)$, denoted by $\tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right)$ with $V_{n,q}$ the Stiefel manifold of $q$-frames in ${\mathbb R}^n$, is the homotopy class of $$x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).$${{endthm}}<br />
<br />
<br />
{{beginthm|Theorem |}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$-immersion. <br />
<br />
If $k<n$ and $\tau\left(f^\prime\right)=0$, then $\left(f,f^\prime\right)$ can be extended to an ${\mathbb R}^q$-immersion $f:D^{k+1}\rightarrow {\mathbb R}^n$. {{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 3.9.<br />
<br />
==References==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hirsch-Smale_theoryHirsch-Smale theory2011-07-05T13:50:35Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the [[h-principle]].<br />
</wikitex><br />
<br />
== Results ==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} For a submanifold $A\subset{\mathbb R}^q$ and a manifold $N$, a pair $\left(f,f^\prime\right)$ is called an ${\mathbb R}^q$-immersion if <br />
<br />
- $f:A\rightarrow N$ is an immersion,<br />
<br />
- $f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$ is a linear map, and<br />
<br />
- there exists an open neighborhood $U$ of $A$ in ${\mathbb R}^q$ and an immersion $g:U\rightarrow N$ such that $g\mid_{A}=f$ and $Dg\mid_{A}=f^\prime$. <br />
{{endthm}}<br />
<br />
{{beginthm|Definition}}Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$-immersion. The obstruction to extending $\tau\left(f,f^\prime\right)$, denoted by $\tau\left(f,f^\prime\right)\in \pi_{k-1}\left(V_{n,q}\right)$ with $V_{n,q}$ the Stiefel manifold of $q$-frames in ${\mathbb R}^n$, is the homotopy class of $$x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).$${{endthm}}<br />
<br />
<br />
{{beginthm|Theorem |}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$-immersion. <br />
<br />
If $k<n$ and $\tau\left(f^\prime\right)=0$, then $\left(f,f^\prime\right)$ can be extended to an ${\mathbb R}^q$-immersion $f:D^{k+1}\rightarrow {\mathbb R}^n$. {{endthm}}<br />
{{cite|Hirsch1959}}, Theorem 3.9.<br />
<br />
==References==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hyperbolic_3-manifoldsHyperbolic 3-manifolds2011-04-01T18:14:27Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
A 3-manifold is hyperbolic if it satisfies the following (equivalent) conditions: <br />
<br>- it admits a complete Riemannian metric of sectional curvature constant -1,<br />
<br><br />
- it admits a Riemannian metric such that its universal covering (with the pull-back metric) is isometric to hyperbolic 3-space ${\Bbb H}^3$,<br />
<br><br />
- it is homeomorphic to $\Gamma\backslash{\Bbb H}^3$, for some discrete, torsion-free group of isometries of hyperbolic 3-space.<br />
<br><br />
<br>Here, hyperbolic 3-space ${\Bbb H}^3$ is the simply connected, complete, Riemannian manifold with sectional curvature constant -1. By Cartans Theorem, ${\Bbb H}^3$ is unique up to isometry. There are different models for ${\Bbb H}^3$, like the upper half-space model, the Poincaré disc model or the hyperboloid model. <br />
<br>The ideal boundary $\partial_\infty{\Bbb H}^3$ can be identified with projective space $P^1{\Bbb C}\cong S^2$. Isometries of hyperbolic 3-space act as conformal automorphisms of the ideal boundary. Thus we can identify the isometry group $Isom\left({\Bbb H}^3\right)$ with the group of conformal automorphisms $Conf\left(S^2\right)$.<br />
<br><br />
<br>The group $SO\left(3,1\right)$ acts on the hyperboloid model and one can use this action to identify $SO\left(3,1\right)$ with the index two subgroup $Isom^+\left({\Bbb H}^3\right)\subset Isom\left({\Bbb H}\right)$ of orientation-preserving isometries. The action is transitive and has $SO\left(3\right)$ as a point stabilizer, thus ${\Bbb H}^3$ is isometric to the homogeneous space $SO\left(3,1\right)/SO\left(3\right)\cong Spin\left(3,1\right)/Spin\left(3\right)$.<br />
<br>The group $PSL\left(2,{\Bbb C}\right)$ acts by fractional-linear automorphisms on $P^1{\Bbb C}$. This action on $\partial_\infty{\Bbb H}^3$ uniquely extends to an action on ${\Bbb H}^3$ by orientation-preserving isometries. One can use this action to identify $PSL\left(2,{\Bbb C}\right)$ with <br />
$Conf^+\left(S^2\right)=Isom^+\left({\Bbb H}^3\right)$.<br />
The action is transitive and has $PSU\left(2\right)$ as a point stabilizer, thus ${\Bbb H}^3$ is isometric to the homogeneous space $PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)\cong SL\left(2,{\Bbb C}\right)/SU\left(2\right)$.<br />
<br><br />
<br>Thus, if $M$ is oriented, then there are two more equivalent conditions:<br />
<br>An oriented 3-manifold is hyperbolic if and only if <br />
<br>- it is homeomorphic to $\Gamma\backslash PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)$ for some discrete, torsion-free subgroup $\Gamma\subset PSL\left(2,{\Bbb C}\right)$,<br />
<br>- <br />
it is homeomorphic to $\Gamma\backslash SO\left(3,1\right)/SO\left(3\right)$ <br />
for some discrete, torsion-free subgroup $\Gamma\subset SO\left(3,1\right)$.<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
By Mostow rigidity, complete hyperbolic metrics of finite volume on a 3-manifold are unique up to isometry. This implies that geometric invariants of the hyperbolic metric, such as the volume and the Chern-Simons-invariant, are topological invariants. <br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<wikitex>;<br />
By the Marden tameness conjecture (proved by Agol and Calegari-Gabai) each hyperbolic 3-manifold with finitely generated fundamental group is the interior of a compact 3-manifold with boundary.<br />
<br><br />
<br>If $M$ is an orientable 3-manifold with boundary, whose interior admits a complete hyperbolic metric of finite volume, then $\partial M$ is a (possibly empty) union of incompressible tori.<br />
<br><br />
<br>Ends of infinite volume fall into two classes, geometrically finite ends and geometrically infinite ends....<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<!-- == Acknowledgments ==<br />
...<br />
<br />
== Footnotes ==<br />
<references/> --><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- == External links ==<br />
* The Wikipedia page about [[Wikipedia:Page_name|link text]]. --><br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Hyperbolic_3-manifoldsHyperbolic 3-manifolds2011-04-01T16:59:05Z<p>Kuessner: Created page with "<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - Fo..."</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
A 3-manifold is hyperbolic if it satisfies the following (equivalent) conditions: <br />
<br>- it admits a complete Riemannian metric of sectional curvature constant -1,<br />
<br><br />
- it admits a Riemannian metric such that its universal covering (with the pull-back metric) is isometric to hyperbolic 3-space ${\Bbb H}^3$,<br />
<br><br />
- it is homeomorphic to $\Gamma\backslash{\Bbb H}^3$, for some discrete, torsion-free group of isometries of hyperbolic 3-space.<br />
<br><br />
<br>Here, hyperbolic 3-space ${\Bbb H}^3$ is the simply connected, complete, Riemannian manifold with sectional curvature constant -1. By Cartans Theorem, ${\Bbb H}^3$ is unique up to isometry. There are different models for ${\Bbb H}^3$, like the upper half-space model, the Poincaré disc model or the hyperboloid model. <br />
<br>The ideal boundary $\partial_\infty{\Bbb H}^3$ can be identified with projective space $P^1{\Bbb C}\cong S^2$. Isometries of hyperbolic 3-space act as conformal automorphisms of the ideal boundary. Thus we can identify the isometry group $Isom\left({\Bbb H}^3\right)$ with the group of conformal automorphisms $Conf\left(S^2\right)$.<br />
<br><br />
<br>The group $SO\left(3,1\right)$ acts on the hyperboloid model and one can use this action to identify $SO\left(3,1\right)$ with the index two subgroup $Isom^+\left({\Bbb H}^3\right)\subset Isom\left({\Bbb H}\right)$ of orientation-preserving isometries. The action is transitive and has $SO\left(3\right)$ as a point stabilizer, thus ${\Bbb H}^3$ is isometric to the homogeneous space $SO\left(3,1\right)/SO\left(3\right)\cong Spin\left(3,1\right)/Spin\left(3\right)$.<br />
<br>The group $PSL\left(2,{\Bbb C}\right)$ acts by fractional-linear automorphisms on $P^1{\Bbb C}$. This action on $\partial_\infty{\Bbb H}^3$ uniquely extends to an action on ${\Bbb H}^3$ by orientation-preserving isometries. One can use this action to identify $PSL\left(2,{\Bbb C}\right)$ with <br />
$Conf^+\left(S^2\right)=Isom^+\left({\Bbb H}^3\right)$.<br />
The action is transitive and has $PSU\left(2\right)$ as a point stabilizer, thus ${\Bbb H}^3$ is isometric to the homogeneous space $PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)\cong SL\left(2,{\Bbb C}\right)/SU\left(2\right)$.<br />
<br><br />
<br>Thus, if $M$ is oriented, then there are two more equivalent conditions:<br />
<br>An oriented 3-manifold is hyperbolic if and only if <br />
<br>- it is homeomorphic to $\Gamma\backslash PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)$ for some discrete, torsion-free subgroup $\Gamma\subset PSL\left(2,{\Bbb C}\right)$,<br />
<br>- <br />
it is homeomorphic to $\Gamma\backslash SO\left(3,1\right)/SO\left(3\right)$ <br />
for some discrete, torsion-free subgroup $\Gamma\subset SO\left(3,1\right)$.<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<!-- == Acknowledgments ==<br />
...<br />
<br />
== Footnotes ==<br />
<references/> --><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- == External links ==<br />
* The Wikipedia page about [[Wikipedia:Page_name|link text]]. --><br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Symplectic_manifoldsSymplectic manifolds2010-12-02T09:40:28Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}== Introduction ==<br />
<wikitex>;<br />
A '''symplectic manifold''' is a smooth manifold $M$ together with a differential two-form $\omega$ that is nondegenerate and closed. The form $\omega$ is called a '''symplectic form'''. The nondegeneracy means that the highest nonzero power of $\omega$ is a volume form on $M.$ It follows that a symplectic manifold is even dimensional.<br />
<br />
Symplectic manifolds originated from classical mechanics. The phase space of a dynamical system is the cotangent bundle of the configuration space and it is equipped with a symplectic form. This symplectic form is preserved by the flow of the system.<br />
</wikitex><br />
== Examples ==<br />
<wikitex><br />
$\bullet$<br />
The most basic example of a symplectic manifold is $\mathbb R^{2n}$ equipped with the form <br />
$\omega_0:=dx^1\wedge dy^1 + \ldots + dx^n\wedge dy^n.$ <br />
<br />
A theorem of Darboux {{cite|McDuff&Salamon1998}} states that locally every <br />
symplectic manifold if of this form. More precisely, if $(M,\omega)$ is a symplectic $2n$-manifold<br />
then for every point $x\in M$ there exists an open neighbourhood $U\subset M$ of $p$ and a <br />
diffeomorphism $f\colon U\to f(U)\subset \mathbb R^{2n}$ such that the restriction of $\omega$<br />
to $U$ is equal to the pull-back $f^*\omega_0.$ This implies that symplectic manifolds have<br />
no local invariants.<br />
<br />
$\bullet$<br />
An area form on an oriented surface is symplectic.<br />
<br />
$\bullet$<br />
Let $X$ be a smooth manifold and let $\lambda$ be a one-form on the cotangent bundle $T^*X$ defined as follows.<br />
If $V$ is a vector tangent to $T^*X$ at a point $\alpha$ then $\lambda_{\alpha}(X) = \alpha (\pi_*(X)),$ where<br />
$\pi\colon T^*X\to X$ is the projection. In local coordinates the form $\lambda$ can be expressed as<br />
$\sum y^idx^i.$ The differential $d\lambda$ is a symplectic form on the cotangent bundle $T^*X.$<br />
<br />
$\bullet$<br />
If $(M,\omega)$ is a closed, i.e. compact and without boundary, symplectic $2n$-manifold then the cohomology classes<br />
$[\omega]^k$ are non-zero for $k=0,1.\ldots,n.$ This follows from the fact that the cohomology class of the volume<br />
form $\omega^n$ is nonzero on a closed manifold. This necessary condition implies that spheres of dimension greater than<br />
two are not symplectic. More generally, no closed manifold of the form $M \times S^k$ is symplectic for $k>2.$<br />
<br />
$\bullet$<br />
The complex projective space $\mathbb C \mathbb P^n$ is symplectic with respect to its Kähler form.<br />
Its pull back to a complex projective smooth manifold $X \subset \mathbb C \mathbb P^n$ is also symplectic.<br />
More generally, every Kähler manifold is symplectic.<br />
</wikitex><br />
<br />
== Symmetries ==<br />
<wikitex>;<br />
A diffeomorphism $f\colon M\to M$ of a symplectic manifold $(M,\omega)$ is called symplectic if it preserves<br />
the symplectic form, $f^*\omega = \omega.$ Sometimes such a diffeomorphism is called a symplectiomorphism.<br />
The group of all symplectic diffeomorphisms of $(M,\omega)$ is denoted by <br />
$\operatorname{Symp}(M,\omega).$<br />
<br />
It follows from the nondegeneracy of the symplectic form $\omega$ the map $X \mapsto \iota_X\omega$ defines an isomorphism<br />
between the vector fields and the one-forms on a symplectic manifold $(M,\omega).$ If the flow of a vector field $X$ <br />
preserves the symplectic form we have that $0 = L_X\omega = d\iota_X \omega + \iota _X d\omega.$ Then the closedness<br />
of the symplectic form implies that the one-form $\iota_X\omega$ is closed. It follows that the Lie algebra of<br />
the group of symplectic diffeomorhism consists of the vector fields $X$ for which the one-form $\iota _X \omega$<br />
is closed. Hence it can be identified with the space of closed one-forms.<br />
<br />
If the one-form $\iota _X \omega$ is exact, i.e. $\iota _X \omega = dH$ for some function $H\colon M\to \mathbb R$<br />
then the vector field $X$ is called Hamiltonian. Symplectic diffeomorphism generated by Hamiltonian flows form<br />
a group $\operatorname{Ham}(M,\omega)$ called the group of Hamiltonian diffeomorphism. Its Lie algebra can be<br />
identified with the quotient of the space of smooth functions on $M$ by the constants.<br />
</wikitex><br />
==Constructions==<br />
===Products=== <br />
<wikitex>;<br />
The product of symplectic manifolds $(M_1,\omega_1)$ and $(M_2,\omega_2)$ is a symplectic manifold with <br />
respect to the form $a\cdot p_1^*\omega_1 + b\cdot p_2^*\omega_2$ for nonzero real numbers <br />
$a,b\in \mathbb R.$ Here $p_i\colon M_1\times M_2\to M_i$ is the projection.<br />
</wikitex><br />
<br />
===Bundles=== <br />
<wikitex>;<br />
A locally trivial bundle $M\to E\to B$ is called symplectic (resp. Hamiltonian) if its structure<br />
group is a subgroup of the group of symplectic (resp. Hamiltonian) diffeomorphisms.<br />
<br />
'''Example.''' The product of the Hopf bundle with the circle is a symplectic bundle $T^2 \to S^3 \times S^1 \to S^2.$<br />
Indeed, the structure group is a group of rotations of the torus and hence it preserves the area. <br />
As we have seen above the product $S^3 \times S^1$ does not admit a symplectic form. This example<br />
shows that, in general, the total space of a symplectic bundle is not symplectic.<br />
<br />
Let $M\stackrel {i}\to E\stackrel{\pi}\to B$ is a compact symplectic bundle over a symplectic base.<br />
According to a theorem of Thurston, if there exists a cohomology class $a\in H^2(E)$ such that<br />
its pull back to every fibre is equal to the class of the symplectic form of the fibre<br />
then there exists a representative $\alpha $ of the class $a$ such that<br />
$\Omega := \alpha + k\cdot \pi^*(\omega_B)$ is a symplectic form on $E$ for every big enough $k.$<br />
<br />
A symplectic fiber bundle may have a symplectic form on the total space which restricts symplectically to the fibers, even if the base is not symplectic. Such bundles are constructed using '''fat connections'''. Let there be given a principal fiber bundle<br />
$$G\rightarrow P\rightarrow M.$$<br />
Let $\theta$ be a connection form, $\Omega$ the curvature form of this connection, and $\mathcal{H}$ be the horizontal distribution. A vector $v\in\frak{g}^*$ is called ''fat'' (with respect to the given connection), if the 2-form <br />
$$(X,Y)\rightarrow \langle \Omega(X,Y),v\rangle$$<br />
is nondegenerate on the horizontal distribution of the connection. The following theorem yields a construction of a symplectic form on the total space of fiber bundles associated with principal bundles equipped with particular connections. <br />
<br />
{{beginthm|Theorem|}} Let there be given a symplectic manifold $(F,\omega)$ endowed with a hamiltonian action of a Lie group $G$. Let $\mu: F\rightarrow\frak{g}^*$ be the moment map of the $G$-action. If $\mu(F)\subset\frak{g}^*$ consists of fat vectors, then the associated bundle <br />
$$F\rightarrow P\times_GF\rightarrow M$$<br />
admits a fiberwise symplectic form on the total space.<br />
{{endthm}}<br />
using this theorem, one can construct examples of symplectic fiber bundles with fiberwise symplectic form on the total space (see examples below).<br />
<br />
'''Example (twistor bundles)'''<br />
Consider the principal bundle of the orthogonal frame bundles over $2n$-dimensional manifold $M$:<br />
$$SO(2n)\rightarrow P\rightarrow M.$$<br />
Let $F=SO(2n)/U(n)$. The associated bundle with fiber $SO(2n)/U(n)$ is called the '''twistor bundle'''. It is easy to see that $SO(2n)/U(n)$ can be identified with a coadjoint orbit $F_{\xi}$ of some $\xi\in\frak{g}^*$, where $\frak{g}^*$ denotes the Lie algebra of $SO(2n)$. Moreover, if $M$ admits Riemannian metric of pinched curvature with sufficiently small pinching constant then $\xi$ is fat with respect to the Levi-Civitta connection in the frame bundle. As a result, the whole coadjoint orbit (which is the image of the moment map of the $SO(2n)$-action) consists of fat vectors. Thus, we obtain a fiberwise symplectic structure on the total space of any twistor bundle <br />
$$F_{\xi}\rightarrow P\times_{SO(2n)}F_{\xi}\rightarrow M$$<br />
over even-dimensional manifolds of pinched curvature. In particular, twistor bundles over spheres $S^{2n}$ or hyperbolic manifolds, admit fiberwise symplectic structures. The simplest example of this construction is the fibering of $\mathbb{C}P^3$ over $S^4$ with fiber $\mathbb{C}P^1$, since it is known that the total space of the twistor bundle over $S^4$ is $\mathbb{C}P^3$. <br />
<br />
'''Example (locally homogeneous complex manifolds)'''<br />
Let $G$ be a Lie group of non-compact type, which is a real form of a complex Lie group $G^c$. Choose a parabolic subgroup $B\subset G^c$ and a maximal compact subgroup $K$ in $G$. Assume that $V=B\subset G$ is compact. Then one can show that $K/V$ can be identified with a coadjoint orbit of some vector in $\frak{k}^*$, which is fat with respect to a $K$-invariant connection in the principal bundle<br />
$$K\rightarrow G/V\rightarrow G/K.$$<br />
It follows that the associated bundle<br />
$$K/V\rightarrow G/V\rightarrow G/K$$<br />
is a symplectic fiber bundle with fiberwise symplectic structure. This construction can be compactified by taking lattices in $G$ which intersect trivially with $K$. A particular example is given by the fiber bundle<br />
$$SO(2n)/U(n)\rightarrow SO(2n,p)/U(n)\times SO(p)\rightarrow SO(2n,p)/SO(2n)\times SO(p),$$<br />
and its compactification by lattices.</wikitex><br />
<br />
===Symplectic reduction===<br />
<wikitex>;<br />
Let $G$ be a Lie group acting on a symplectic manifold $(M,\omega)$ in a hamiltonian way. Denote <br />
by $\mu: M\rightarrow\frak{g}^*$ the moment map of this action. Since $G$ acts on the level set <br />
$\mu^{-1}(a),a\in\frak{g}^*$, one can consider the orbit space $\mu^{-1}(a)/G$. It is an orbifold in general, but it happens to be a manifold, when $G$ acts freely on the preimage, and $a$ is a regular point. In this case, $\tilde M=\mu^{-1}(a)/G$ is a symplectic manifold as well, called '''symplectic reduction'''. It is often denoted by $M//G$.<br />
</wikitex><br />
===Symplectic cut===<br />
<wikitex>;<br />
Let $(M,\omega)$ be a symplectic manifold with a hamiltonian action of the circle $S^1.$ If $\mu:\rightarrow \mathbb R$ is the moment map, $M_a=\mu^{-1}(a)$ is a regular level, then the action restricted to $M_a$ has no fixed points, hence $M_a$ is the boundary of the associated disk bundle W. This is a manifold if the action is free and an orbifold if a non-trivial isotropy occurs.<br />
</wikitex><br />
===Coadjoint orbits===<br />
<br />
===Symplectic homogeneous spaces===<br />
<br />
Nilmanifolds, solvmanifolds, homogeneous spaces of semisimple Lie groups<br />
<br />
===Donaldson's theorem on submanifolds===<br />
<br />
===Surgery in codimension 2===<br />
<wikitex>;<br />
Consider two symplectic manifolds $(M_1,\omega_1),(M_2,\omega_2)$ of equal dimension and suppose that there are codimension two symplectic submanifolds $V_1\subset M_1,V_2\subset M_2$ and a symplectomorphism $f:V_1\rightarrow V_2$ such that Chern classes of normal bundles satisfy $f^*c_1(\nu_2)=-c_1(\nu_1).$ <br />
Then by removing tubular neighborhods of $V_1$ and $V_2$ we get manifolds with boundaries. The map $f$ induces a diffeomorphism of the boundaries, one can form a new manifold identifying the boundaries by this diffeomorphism and define on it a symplectic form which coincides with <br />
$\omega_1$ and $\omega_2$ outside of a tubular neigborhood of the trace of glueing {{cite|Gompf1995}}. The same works if $V_1,V_2$ are symplectic submanifolds of a connected symplectic manifold. </wikitex><br />
<br />
===Symplectic blow-up===<br />
<br />
== Invariants ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Rho-invariantRho-invariant2010-12-02T09:35:13Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
<br />
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<br />
To achieve a unified layout, along with using the template below,<br />
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use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
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<br />
== Introduction == <br />
<wikitex>; <br />
The $\rho$-invariant is an invariant of odd-dimensional closed manifolds closely related to the<br />
equivariant signature. Its definition is motivated by the<br />
equivariant signature defect of even-dimensional manifolds with<br />
boundary. Namely, for manifolds with boundary the classical index<br />
formula for equivariant signature must be corrected by an additional<br />
term. It is this term which gives rise to the $\rho$-invariant.<br />
There is also another definition using bordism theory. Finally there<br />
is also an analytic defintion as a relative $\eta$-invariant.<br />
</wikitex><br />
<br />
== Background ==<br />
<br />
=== G-index theorem ===<br />
<wikitex>; <br />
Let $G$ be a compact Lie group acting smoothly on a smooth manifold $Y^{2d}$. The middle intersection form becomes a non-degenerate $(-1)^d$-symmetric bilinear form on which<br />
$G$ acts. The positive and negative definite subspaces are $G$-invariant and hence such a form yields an element in the representation ring $R(G)$ denoted by $\mathrm{sign_G} (Y)$.<br />
In fact $\mathrm{sign_G} (Y) \in R^{(-1)^d} (G)$ which in terms of characters means that we obtain a real (case $d$ even) / purely imaginary (case $d$ odd) character, which will be denoted as <br />
$$<br />
\mathrm{sign_G} (-,Y) \colon g \in G \mapsto \mathrm{sign_G} (g,Y) \in \Cc.<br />
$$<br />
<br />
The (cohomological version of the) Atiyah-Singer $G$-index theorem {{cite|Atiyah&Singer1968b|Theorem (6.12)}} tells us that if $Y$ is closed then for all $g \in G$<br />
$$ <br />
\mathrm{sign_G} (g,Y) = L(g,Y) \in \Cc,<br />
$$<br />
where $L(g,Y)$ is an expression obtained by evaluating certain cohomological classes on the fundamental classes of the $g$-fixed point submanifolds $Y^g$ of $Y$. In particular if the action is free then $\mathrm{sign_G} (g,Y) = 0$ if $g \neq 1$. This means that $\mathrm{sign_G} (Y)$<br />
is a multiple of the regular representation. This theorem was generalized by Wall to topological semifree actions on topological manifolds, which is the case we will need here<br />
{{cite|Wall1999|chapter 14B}}.<br />
<br />
The assumption that $Y$ is closed is essential, and motivates the definition of the $\rho$-invariant. <br />
</wikitex><br />
<br />
=== Cobordism theory ===<br />
<wikitex>; <br />
Also the result from cobordism theory is needed which says that for an odd-dimensional manifold $X$ with a finite fundamental group $G$ there always exists a $k \in \Nn$ and a manifold with boundary $(Y,\partial Y)$ such that $\pi_1 (Y) \cong \pi_1 (X) \cong G$ and $\partial Y = k \cdot X$. In other words <br />
$$<br />
\Omega^{\textup{STOP}}_{2d-1} (BG) \otimes \Qq = 0 <br />
$$<br />
This is due to {{cite|Conner&Floyd1964}}, {{cite|Williamson1966}} and {{cite|Madsen&Milgram1979}}.<br />
</wikitex><br />
<br />
== Definition == <br />
<br />
=== G finite ===<br />
<wikitex>; <br />
<br />
{{beginthm|Definition}} \label{defn-rho-1} <br />
<br />
Let $X^{2d-1}$ be a closed manifold with $\pi_1 (X) \cong G$ a finite group. Define<br />
$$<br />
\rho (X) = \frac{1}{k} \cdot \mathrm{sign_G} (\widetilde Y) \in \Qq R^{(-1)^d}<br />
(G)/ \langle \mathrm{reg} \rangle<br />
$$<br />
for some $k \in \Nn$ and $(Y,\partial Y)$ such that $\pi_1 (Y) \cong<br />
\pi_1 (X)$ and $\partial Y = k \cdot X$. The symbol $\langle<br />
\textup{reg} \rangle$ denotes the ideal generated by the regular<br />
representation. <br />
{{endthm}}<br />
<br />
See {{cite|Atiyah&Singer1968b|Remark after Corollary 7.5}} for more details. Note that the manifold $Y$ in the definition always exists by the above mentioned result in cobodism theory. Furthermore the invariant is well-defined thanks to the cohomological version of the $G$-index theorem.<br />
</wikitex><br />
<br />
=== G compact Lie group ===<br />
<wikitex>; <br />
{{beginthm|Definition|(Atiyah-Singer)}} \label{defn-rho-2}<br />
Let $G$ be a compact Lie group acting freely on a manifold $\widetilde{X}^{2d-1}$. Suppose in addition that there is a manifold with boundary $(Y,\partial Y)$ on which $G$ acts (not necessarily freely) and such that $\partial Y = \widetilde X$. Define<br />
$$<br />
\rho_G (\widetilde X) \co g \in G \mapsto \mathrm{sign_G} (g,Y) - L(g,Y) \in \Cc.<br />
$$<br />
{{endthm}}<br />
<br />
See {{cite|Atiyah&Singer1968b|Theorem 7.4}} for more details.<br />
<br />
In this definition we think about the $\rho$-invariant as about a function $G \smallsetminus \{1\} \rightarrow \Cc$. <br />
<br />
When both definitions apply (that means when $G$ is a finite group), then they coincide, that means $\rho (X) = \rho_G (\widetilde X)$.<br />
</wikitex><br />
<br />
<br />
== References == {{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Links_of_singular_points_of_complex_hypersurfacesLinks of singular points of complex hypersurfaces2010-06-21T09:52:58Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
The links of singular points of complex hypersurfaces provide a large class of examples of highly-connected odd dimensional manifolds. A standard reference is {{cite|Milnor1968}}. See also {{cite|Hirzebruch&Mayer1968}} and {{cite|Dimca1992}}.<br />
<br />
These manifolds are the boundaries of [[highly-connected]], [[stably parallelisable]] even dimensional manifolds and hence stably parallelisable themselves. In the case of singular points of complex curves, the link of such a singular point is a [[fibered link]] in the $3$-dimensional sphere $S^3$. <br />
</wikitex><br />
<br />
== Construction and properties==<br />
<wikitex>;<br />
Let $f(z_1, \dots, z_{n+1})$ be a non-constant polynomial in $n+1$ complex variables. A complex hypersurface $V$ is the algebraic set consisting of points $z=(z_1, \dots, z_{n+1})$ such that $f(z)=0$. A regular point $z \in V$ is a point at which some partial derivative $\partial f /\partial z_j$ does not vanish; if at a point $z \in V$ all the partial derivatives $\partial f / \partial z_j$ vanish, $z$ is called a '''singular point''' of $V$. <br />
<br />
Near a regular point $z$, the complex hypersurface $V$ is a smooth manifold of real dimension $2n$; in a small neighborhood of a singular point $z$, the topology of the complex hypersurface $V$ is more complicated. One way to look at the topology near $z$, due to Brauner, is to look at the intersection of $V$ with a $(2n+1)$-dimensioanl sphere of small radius $\epsilon$ $S_{\epsilon}$ centered at $z$. <br />
<br />
\begin{thm}<br />
The space $K=V\cap S_{\epsilon}$ is $(n-2)$-connected.<br />
\end{thm}<br />
<br />
The homeomorphism type of $K$ is independent of the small parameter $\epsilon$, it is called the '''link''' of the singular point $z$.<br />
<br />
\begin{thm}(Fibration Theorem)\label{fibration}<br />
For $\epsilon$ sufficiently small, the space $S_{\epsilon}-K$ is a smooth fiber bundle over $S^1$, with projection map $\phi \colon S_{\epsilon}-K \to S^1$, $z \mapsto f(z)/|f(z)|$. Each fiber $F_{\theta}$ is parallelizable and has the homotopy type of a finite CW-complex of dimension $n$.<br />
\end{thm}<br />
<br />
The fiber $F_{\theta}$ is usually called the '''Milnor fiber''' of the singular point $z$.<br />
<br />
A singular point $z$ is '''isolated''' if there is no other singular point in some small neighborhood of $z$.<br />
<br />
In this special situation, the above theorems are strengthened to the following<br />
<br />
\begin{thm}<br />
Each fiber $F_{\theta}$ is a smooth parallelizable manifold, the closure $\overline{F_{\theta}}$ has boundary $K$ and the homotopy type of a bouquet of $n$-spheres $S^n\vee \cdots \vee S^n$.<br />
\end{thm}<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
Seen from the above section, the link $K$ of an isolated singular point $z$ of a complex hypersurface $V$ of complex dimension $n$ is an $(n-2)$-connected $(2n-1)$-dimensional closed smooth manifold. In high dimensional topology, these are called [[highly-connected|highly connected manifolds]], since for a $(2n-1)$-dimensional closed manifold $M$ which is not a homotopy sphere, $(n-2)$ is the highest connectivity $M$ could have. Therefore to understand the classification and invariants of the links $K$ one needs to understand the classification and invariants of [[highly-connected]] odd dimensional manifolds.<br />
<br />
On the other hand, as the link $K$ is closely related to the singular point $z$ of the complex hypersurfaces, some of the topological invariants of $K$ are computable from the polynomial. <br />
<br />
Let $V$ be a complex hypersurface defined by $f(z_1, \dots, z_{n+1})$, $z^0$ be an isolated singular point of $V$. Let $g_j=\partial f /\partial z_j$ $j=1, \dots, n+1$. By putting all these $g_j$'s together we get the gradient field of $f$, which can be viewed as a map $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$, $z \mapsto (g_1(z), \dots, g_{n+1}(z))$. If $z^0$ is an isolated singular point, then $z \mapsto g(z)/||g(z)||$ is a well-defined map from a small sphere $S_{\epsilon}$ centered at $z^0$ to the unit sphere $S^{2n+1}$ of $\mathbb C^{n+1}$. The mapping degree $\mu$ is called the multiplicity of the isolated singular point $z^0$. ($\mu$ is also called the Milnor number of $z^0$.)<br />
<br />
\begin{thm}<br />
The middle homology group $H_n(F_{\theta})$ is a free abelian group of rank $\mu$.<br />
\end{thm}<br />
<br />
Furthermore, the homology groups of the link $K$ are determined from the long homology exact sequence<br />
$$ \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0$$<br />
of the pair $(\overline{F_{\theta}},K)$. The map $j_*$ is the adjoint of the intersection pairing on $\overline{F_{\theta}}$ <br />
$$s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.$$<br />
A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. $s$ is called the '''Milnor lattice''' of the singular point. Thus the homology groups of the link $K$ is completely determined by the Milnor lattice of the singular point.<br />
<br />
== Topological spheres as links of singular points ==<br />
<wikitex>;<br />
Especially, the link $K$ is an integral [[homology sphere]] if and only if the intersection form $s$ is unimodular, i.~e.~the matrix of $s$ has determinant $\pm1$. If $n\ge 3$, the [[Generalized Poincare Conjecture]] implies that $K$ is a topological sphere. <br />
<br />
By Theorem \ref{fibration}, there is a smooth fiber bundle over $S^1$ with fiber $F_{\theta}$. The natural action of a generator of $\pi_1(S^1)$ induces the characteristic homeomorphism $h$ of the fiber $F_0=\phi^{-1}$. Let $h_* \colon H_n(F_0) \to H_n(F_0)$ be the induced isomorphism on homology and $\Delta(t)=\det(tI-h_*)$ be the '''characteristic polynomial''' of the linear transformation $h_*$. It's a consequence of the [[Wang sequence]] associated with the fiber bundle over $S^1$ that<br />
<br />
\begin{lemma}<br />
For $n \ne 2$ the manifolds $K$ is a topological sphere is and only if the integer $\Delta(1)=\det(I-h_*)$ equals to $\pm 1$.<br />
\end{lemma}<br />
<br />
When $K$ is a topological sphere, as it is the boundary of an $(n-1)$-connected parallelisable $2n$-manifold $\overline{F_0}$, our knowledge of [[exotic spheres]] allows us to determine the diffeomorphism class of $K$ completely:<br />
<br />
* if $n$ is even, the diffeomorphism class of $K$ is determined by the [[Intersection forms#Algebraic invariants|signature]] of the intersection pairing<br />
$$s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z$$<br />
* if $n$ is odd, the diffeomorphism class of $K$ is determined by the [[wikipedia:Kervaire invariant|Kervaire invariant]] <br />
$$c(\overline{F_0}) \in \mathbb Z_2$$<br />
which was computed in {{cite|Levine1966}}<br />
$$c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8$$<br />
$$c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8$$<br />
</wikitex><br />
<br />
== Examples ==<br />
<wikitex>;<br />
The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form <br />
$$f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}$$<br />
where $a_1, \dots, a_{n+1}$ are integers $\ge 2$. The origin is an isolated singular point of $f$.<br />
<br />
\begin{thm}(Brieskorn-Pham)<br />
The group $H_n(F_{\theta})$ is free abelian of rank <br />
$$\mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.)$$<br />
The characteristic polynomial is <br />
$$\Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}),$$<br />
where each $\omega_j$ ranges over all $a_j$-th root of unit other than $1$.<br />
\end{thm}<br />
<br />
The link $K$ is called a [[Exotic spheres#Brieskorn varieties|Brieskorn variety]]. <br />
<br />
For $a_1=\cdots=a_{n+1}=2$, it's seen from the defining equations that the link $K$ is the sphere bundle of the tangent bundle of the $n$-sphere, i.~e.~the [[wikipedia:Stiefel manifold|Stiefel manifold]] $V_{n+1,2}(\mathbb R)$.<br />
<br />
The simplest nontrivial example is $a_1=\cdots =a_n=2$, $a_{n+1}=3$. Then $\omega_1 = \cdots \omega_n=-1$, $\omega_{n+1}=(-1\pm \sqrt{3})/2$. The characteristic polynomial is <br />
$$\Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}$$<br />
$$\Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.$$<br />
For $n=2k+1$ we have $\Delta(1)=1$ so the link $K$ is a topological sphere of dimension $4k+1$; $\Delta(-1)=3$, thus by {{cite|Levine1966}} $K$ has nontrivial [[wikipedia:Kervaire invariant|Kervaire invariant]].<br />
Especially for $k=2$ $K^9$ is the [[Exotic spheres#Plumbing|Kervaire sphere]]. <br />
<br />
The above example is a special case of the $A_k$-singularities, whose defining polynomial is <br />
$$f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}$$<br />
$k$ being an integer $\ge 1$. The Milnor lattice of an $A_k$-singularity is represented by the Dynkin diagram of the simple Lie algebra $A_k$. When $n=3$, the diffeomorphism classification of the link $K$ is obtained from its Milnor lattice and the classification of [[5-manifolds: 1-connected|simply-connected 5-manifolds]]:<br />
* $K$ is diffeomorphic to $S^2 \times S^3$ if $k$ is odd;<br />
* $K$ is diffeomorphic to $S^5$ if $k$ is even.<br />
In this dimension, the diffeomorphism classification of the link $K$ of other types of singular points can be obtained in the same way, once we know the [[Links of singular points of complex hypersurfaces#Invariants|Milnor lattice]] of the singular point. <br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
The link $K$ of a singular point $z$ is the intersection of the hypersurface $V$ defined by $f$ and the sphere $S_{\epsilon}$ in the ambient space $\mathbb C^{n+1}$. Therefore it inherits certain symmetries from that of the singular point. As an example, consider an $A_k$-singularity in $\mathbb C^4$. There is an orientation preserving involution <br />
$$\tau_k \colon \mathbb C^4 \to \mathbb C^4, \ \ (z_1, z_2, _3, z_4) \mapsto (-z_1, -z_2, -z_3, z_4).$$<br />
$\tau_k$ induces an orientation preserving free involution of $K\cong S^5$ or $S^2 \times S^3$. For $k=0,2,4,6$, $\tau_k$'s provide all the 4 smooth free involutions on $S^5$ (see {{cite|Geiges&Thomas1998}}). <br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/3-manifolds3-manifolds2010-06-18T17:30:05Z<p>Kuessner: Undo revision 4022 by Kuessner (Talk)</p>
<hr />
<div>{{Stub}}<br />
<br />
== Introduction ==<br />
<wikitex>;<br />
In the 3-dimensional setting there is no distinction between smooth, PL and topological manifolds neccesary; the categories of smooth, PL and topological manifolds are equivalent (TODO ref). A lot of techniques have been developed in the last century to study 3-manifolds but most of them are very special and don't generalise to higher dimensions. One key idea is to decompose manifolds along incompressible surfaces into smaller pieces, to which certain geometric models apply. <br />
A great progress was made in with the proof of the Poincaré conjecture and Thurton's geometrization conjecture by Perelman in 2003.<br />
[[Image:Phomsphere.jpg|thumb|150px| The universal cover of the famous [[Poincaré homology sphere]] is $S^3$ - here a view of the induced tesselation]]<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<br />
<wikitex>;<br />
Basic examples are $\mathbb{R}^3, S^3, S^1 \times S$ with $S$ any surface. <br />
Important types of 3-manifolds are [[Wikipedia:Haken_manifold|Haken-Manifolds]], [[Wikipedia:Seifert_fibre_spaces|Seifert-Manifolds]], [[Wikipedia:Lens_spaces|3-dimensional lens spaces]], [[Wikipedia:Torus_bundle|Torus-bundles and Torus semi-bundles]].<br />
<br />
There are two topological processes to join 3-manifolds to get a new one.<br />
The first is the connected sum of two manifolds $M_1$ and $M_2$. Choose embeddings $f_1:D^3\rightarrow M_1$ and $f_2:D^3\rightarrow M_2$, remove the interior of $f_1(D^3)$ and $f_2(D^3)$ and glue $M_1$ and $M_2$ together along the boundaries $f_1(S^3)$ and $f_2(S^3)$.<br />
The second uses incompressible surfaces. Let $M$ be manifold and $S\subset M$ a surface. $S$ is incompressible, if there is no disk $D$ in $M$ with $D\cap S=\partial D$.<br />
The torus sum is the process which glues incompressible tori boundary components together.<br />
<br />
(TODO What is incompressibility needed? / What is is good for/ What happen if one takes a compressible surface ?)<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
In the 3-dimensional world the fundamental group is a powerful invariant to distiguish manifolds.<br />
It determines already all homology groups: <br />
* $H_1(M)$ = abelization of $\pi_1(M)$. <br />
* $H_2(M) = H^1(M) = H_1(M)/$torsion<br />
* $H_3(M) = \Zz$<br />
* $H_n(M) = 0$ for $n > 3$ <br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
By reversing the process of connected and torus sum every 3-manifold can be decomposed into pieces which admit a geometric structure. We describe the details in the following.<br />
<br />
=== Prime decomposition === <br />
<wikitex>;<br />
{{beginthm|Definition}}<br />
A manifold $M$ is called prime, if it can't be written as a non-trivial connected sum, i.e. $M=M_1 \# M_2$ implies $M_1 = S^3$ or $M_2 = S^3$.<br />
A manifold $M$ is called irreducible if every embedded $S^2$ bounds a ball, i.e. the embedding extends to an embedding of $D^3$<br />
{{endthm}}<br />
Irreducibility is only slightly stronger than being prime. A orientable prime 3-manifold is either $S^2 \times S^1$ or every embedded 2-sphere bounds a ball.<br />
<br />
\begin{theorem}[Kneser]<br />
Every orientable, compact 3-manifold $M$ has a decomposition $M=P_1 \# \ldots \# P_n$ into prime manifolds $P_i$ unique up to ordering and $S^3$ summands.<br />
\end{theorem} <br />
<br />
A orientable prime 3-manifold is either $S^2 \times S^1$ or every embedded 2-sphere bounds a ball, in which case the manifold is called irreducible.<br />
<br />
Van Kampen's theorem tells you, that $\pi_1(M \# N)=\pi_1(M)*\pi_1(N)$. Hence any 3-manifold, whose fundamental group cannot be written as a free product of two nontrivial subgroups, can only be written as the connected sum of another 3-manifold with a simply connected 3-manifold. By the Poincaré conjecture a simply connected 3-manifold is already homeomorphic to $S^3$. Hence each such manifold is prime.<br />
<br />
Prime 3-manifolds can be distinguished by their fundamental groups into the following 3 types:<br />
</wikitex><br />
==== Type I: finite fundamental group==== <br />
<wikitex>;<br />
The universal cover $\tilde{M}$ is a simply-connected 3-manifold. As the fundamental group already determines the homology of a oriented, closed compact 3-manifold, it has to be a homology sphere. Using the Hurewicz-theorem, its fundamental class is represented by a degree 1 map $S^3 \rightarrow \tilde{M}$. This map induces isomorphisms on the homology and on the fundamental group. Hence it is a weak homotopy equivalence, and hence a homotopy equivalence by Whitehead's theorem (ref?).<br />
Hence every prime $3$-manifold with finite fundamental group arises as the quotient of a homotopy sphere by a free action of a finite group.<br />
With the use of the Poincaré conjecture every homotopy 3-sphere is homeomorphic to $S^3$ and we can write $M=S^3/\Gamma$. If $\Gamma$ is cyclic $M$ is known as lens space (ref). <br />
</wikitex><br />
==== Type II: infinite cyclic fundamental group ====<br />
<wikitex>;<br />
$S^1\times S^2$ is the only orientable closed prime 3-manifold of this type. Futhermore it is the only not irreducible prime manifold. (TODO: proof/ref)<br />
</wikitex><br />
<br />
==== Type III: infinite non-cyclic fundamental group ====<br />
<wikitex>;<br />
Such a manifold $M$ is always aspherical (TODO ref). The sphere theorem states, that every map $S^2\rightarrow M$ is homotopic to an embedding; and - as $M$ is irreducible - it is nullhomotopic. Hence $\pi_2(M)=0$. Consider the universal covering $\tilde{M}$ of $M$. Its first homology vanishes as it is simply connected. The long exact sequence of homotopy groups of the fibration $pi_1(M)\rightarrow \tilde{M}\rightarrow M$ gives a isomorphism $pi_2(M)\cong \pi_2(\tilde{M})$. Hence by Hurewicz' theorem $H_2(\tilde{M})=0$. Furthermore $H_3(\tilde{M})=0$, as $M$ is noncompact. Applying Hurewicz theorem again we get that all homotopy groups of $\tilde{M}$ vanish and hence by Whitehead's theorem $\tilde{M}$ is contractible. This means that $M$ is apherical. <br />
Hence the homotopy type of a prime 3-manifold with infinite non-cyclic fundamental group is uniquely determined by its fundamental group.<br />
Furthermore not every group can occur as a fundamental group of a prime 3-manifold. The equivariant cellular chain complex of $\tilde{M}$ is a projective resolution of the trivial $\Zz[\pi_1(M)]$-module $\Zz$. Hence ....<br />
For any subgroup $F\le \pi_1(M)$ the space $\tilde{M}/F$ is a finite-dimensional model for $K(F,1)$. For example a finite group cannot have such a model (by group homology ref) and hence $\pi_1(M)$ must be torsionfree. Furthermore it is a Poincaré duality group (link).<br />
</wikitex><br />
<br />
=== Torus decomposition ===<br />
<wikitex>;<br />
According to the previous section it remains to classify irreducible prime 3-manifolds. <br />
After cutting along spheres which don't bound balls as far as possible the next canonical step is to consider incompressible tori which are disjoint from the boundary. <br />
<br />
\begin{theorem}[Jacob-Shalen, Johannson]<br />
If $M$ is an irreducible compact orientable manifold, then there is a collection of disjoint incompressible tori $T_1, \ldots ,T_n$ in $M$ such that splitting $M$ along the union of these tori produces manifolds $M_i$ which are either [[Wikipedia:Seifert_fiber_space|Seifert-fibered]] or atoroidal, i.e. every incompressible torus in $M_i$ is isotopic to a torus component of $\partial M_i$. Furthermore, a minimal such collection of tori $T_j$ is unique up to isotopy in $M$.<br />
\end{theorem}<br />
<br />
Thurston's geometrization conjectures states that all the pieces we get by this JSJ-decomposition admit one of eight possible geometric structures:<br />
There is a list of eight simply connected Riemannian manifolds - the so called model geometries. A geometric structure on $M$ is the choice of a Riemannian metric on $M$, with the property that its universal covering $ \tilde{M}$ equipped with the pull-back metric is isometric to one of the eight model geometries. It might a priori be easier to classify all cocompact actions on the several model geometries.<br />
<br />
The Seifert-fibered pieces are well understood since the work of Seifert in the 30s (TODO: mention classification theorem). The atoroidal pieces are described by the following Hyperbolization theorem which was stated by Thurston (ref) and proven by Perelman.<br />
\begin{thm}<br />
Every irreducible atoroidal closed 3-manifold that is not Seifert-fibred is hyperbolic.<br />
\end{thm}<br />
</wikitex><br />
<br />
=== Dehn surgery ===<br />
<wikitex><br />
Dehn surgery is a way of constructing (TODO oriented ? neccesary) 3-manifolds. Given a [[Wikipedia:Link_%28knot_theory%29 | link]] in a $3$-manifold $N$<br />
$$L: \coprod_{i=1}^n S^1\rightarrow N,$$<br />
and a choice of a tubular neighborhood of $L$ <br />
$$L': \coprod_{i=1}^n S^1\times D^2\rightarrow N\mbox{ with }L'(x,0)=L(x)$$.<br />
(This choice essentially is the choice of a trivialization of the normal bundle; TODO find a correct formulation for this).<br />
This gives us a family of embedded, disjoint, full tori. The idea of Dehn surgery is to remove these Tori and glue them back in using a twist.\\<br />
Let us restrict to the case with only one solid torus $L':S^1\times D^2\rightarrow N$. <br />
Choose any self-homeomorphism $f$ of the torus $S^1\times S^1$. The result of the Dehn surgery at $L$ with the twist $f$ is defined as <br />
$$N_{f,L'}:=N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N\setminus L'(S^1\times \mathring{D}^2) \amalg S^1\times D^2/\sim,$$<br />
where the equivalence relation identifies for $(x,y)\in S^1\times S^1$ the points $L'(x,y)$ in the left component and $f(x,y)$ in the right component.<br />
If $f$ is the coordinate flipping, Dehn surgery is nothing but usual codimension $2$ surgery.<br />
<br />
\begin{lemma}<br />
Suppose $f,f'\in \Homeo(T^2)$ are isotopic and let $L':S^1\times D^2 \rightarrow N$ be any embedding of the full Torus in a $3$-Manifold $N$.Then $N_{f,L'}$ and $N_{f',L'}$ are homeomorphic.<br />
\end{lemma}<br />
\begin{proof}<br />
Let $j:T^2\times [0;1] \rightarrow T^2$ be an isotopy from $f$ to $f'$. This gives a homeomorphism:<br />
$$ J:T^2\times [0;1]\rightarrow T^2\times [0;1] \qquad (x,y,t)\mapsto (j(x,y,t),t). $$<br />
(TODO: is its inverse $(x,y,t)\mapsto (j(-,-,t)^{-1}(x,y),t)$ continuous ?).<br />
The idea is to grab some additional space, where one can use the map $J$. TODO<br />
\end{proof}<br />
TODO formulate a lemma, that M_{f,L'} also only depends on the isotopy class of $L'$ (which is hopefully true).<br />
Hence, we have to classify all self-homeomorphisms of $T^2$ up to isotopy.<br />
\begin{lemma}<br />
Every self-homeomorphism of $T^2$ is isotopic to exactly one homeomorphism of the shape <br />
$$f_A:\Rr^2/\Zz^2 \rightarrow \Rr^2 /\Zz^2 \qquad \left(\begin{array}{c}x\\y\end{array}\right)\mapsto A\cdot\left(\begin{array}{c}x\\y\end{array}\right),$$<br />
where $A\in GL_2(\Zz)$ (reference of proof). <br />
\end{lemma}<br />
\begin{proof} Since the torus is a $K\left(\Zz^2,1\right)$-space, we have that $\pi_0 Map\left(T^2,T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is an isomorphism. Homotopic surface homeomorphisms are isotopic (Reference?). Thus the restriction $\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is injective. Moreover, each $A\in Hom\left(\Zz^2,\Zz^2\right)$ is realised by $f_A$, therefore $\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is also surjective. <br />
TODO find a reference in ANY source about the mapping class group<br />
\end{proof}<br />
The next lemma tells us, that composition of self-homeomorphisms corresponds to two successive Dehn surgeries.<br />
\begin{lemma}<br />
Let $f,g\in \Homeo(T^2)$ be given and let $L':S^1\times D^2\rightarrow N$ is an embedding of the full torus in a 3-manifold. Then we have map <br />
$$L'':S^1 \times D^2 \rightarrow N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N_{f,L'}$$<br />
given by the map $S^1\times D^2\rightarrow S^1 \times D^2 \quad (x,y)(x,y/2)$ postcomposed with the canonical inclusion in the second coordinate.<br />
Then $(N_{f,L'})_{g,L''} \cong N_{f\circ g,L'}$. TODO right order of composition ? We will see in the proof.<br />
\end{lemma}<br />
\beg{proof}<br />
\end{proof}<br />
<br />
We have to find out, which self-homeomorphisms of the torus don't change the homeomorphism type of the manifold.<br />
\begin{lemma} Consider a matrix of the form $\left( \begin{array}{cc} 1 & 0 \\ k & 1 \end{array}\right)$ and let $L':S^1\times D^2 \rightarrow N$ be any eembedding. Then $N_{f_A,L'}\cong N$.<br />
\end{lemma}<br />
TODO are there any orientation reversing homeos, that also extend ? Think so. Also add them here.<br />
\begin{proof}<br />
The homeomorphism $f_A \in \Homeo(T^2)$ extends to a homeomorphism of $\bar{f_A}\in \Homeo(S^1\times D^2)$:<br />
$$\bar{f_A}(x,y):=(x,x^ky), $$<br />
where $x\in S^1 = \{z\in \Cc| |z|=1\}, y\in D^2=\{y\in\Cc||y|\le 1\}$.<br />
Using this homeomorphism one can define a homeomorphism from $N_f$ to $N$:<br />
$$N = N\setminus L(S^1\times \mathring{D}^2)\cup_1 S^1\times D^2 \rightarrow N_{f_A}=N\setminus L(S^1 \times \mathring{D}^2)\cup_{f_A} S^1\times D^2$$<br />
given by the identity on the left component and $\bar{f_A}$ on the right component.<br />
\end{proof}<br />
<br />
Together with (link to comment about composition), this tells us, that $N_{f_A}$ really only depends on the coset $A\cdot \left(\begin{array}{cc}1&*\\0&1\end{array}\right)$ (TODO check right or left coset). This coset is uniquely determined by the image $(p,q)$ of $(1,0)$ with $p$ and $q$ coprime.<br />
<br />
The ratio $p/q$ is called the surgery coefficient. (TODO what is the quotient good for ?)<++><br />
<br />
<br />
TODO does the result give different manifolds.<br />
<br />
TODO does the result only depend on the isotopy class of the link. <br />
<br />
Every compact (oriented /able, neccesary ?) 3-manifold might be obtained from $S^3$ by a Dehn surgery along a link (TODO ref). Of course this does not satisfy to classify 3-manifolds without having a good classification of links in $S^3$. <br />
<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
\cite{Scott1983}, \cite{Thurston1997}, \cite{Hatcher2000}, \cite{Hempel1976}<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/3-manifolds3-manifolds2010-06-18T17:26:53Z<p>Kuessner: /* Dehn surgery */</p>
<hr />
<div>{{Stub}}<br />
<br />
== Introduction ==<br />
<wikitex>;<br />
In the 3-dimensional setting there is no distinction between smooth, PL and topological manifolds neccesary; the categories of smooth, PL and topological manifolds are equivalent (TODO ref). A lot of techniques have been developed in the last century to study 3-manifolds but most of them are very special and don't generalise to higher dimensions. One key idea is to decompose manifolds along incompressible surfaces into smaller pieces, to which certain geometric models apply. <br />
A great progress was made in with the proof of the Poincaré conjecture and Thurton's geometrization conjecture by Perelman in 2003.<br />
[[Image:Phomsphere.jpg|thumb|150px| The universal cover of the famous [[Poincaré homology sphere]] is $S^3$ - here a view of the induced tesselation]]<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<br />
<wikitex>;<br />
Basic examples are $\mathbb{R}^3, S^3, S^1 \times S$ with $S$ any surface. <br />
Important types of 3-manifolds are [[Wikipedia:Haken_manifold|Haken-Manifolds]], [[Wikipedia:Seifert_fibre_spaces|Seifert-Manifolds]], [[Wikipedia:Lens_spaces|3-dimensional lens spaces]], [[Wikipedia:Torus_bundle|Torus-bundles and Torus semi-bundles]].<br />
<br />
There are two topological processes to join 3-manifolds to get a new one.<br />
The first is the connected sum of two manifolds $M_1$ and $M_2$. Choose embeddings $f_1:D^3\rightarrow M_1$ and $f_2:D^3\rightarrow M_2$, remove the interior of $f_1(D^3)$ and $f_2(D^3)$ and glue $M_1$ and $M_2$ together along the boundaries $f_1(S^3)$ and $f_2(S^3)$.<br />
The second uses incompressible surfaces. Let $M$ be manifold and $S\subset M$ a surface. $S$ is incompressible, if there is no disk $D$ in $M$ with $D\cap S=\partial D$.<br />
The torus sum is the process which glues incompressible tori boundary components together.<br />
<br />
(TODO What is incompressibility needed? / What is is good for/ What happen if one takes a compressible surface ?)<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
In the 3-dimensional world the fundamental group is a powerful invariant to distiguish manifolds.<br />
It determines already all homology groups: <br />
* $H_1(M)$ = abelization of $\pi_1(M)$. <br />
* $H_2(M) = H^1(M) = H_1(M)/$torsion<br />
* $H_3(M) = \Zz$<br />
* $H_n(M) = 0$ for $n > 3$ <br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
By reversing the process of connected and torus sum every 3-manifold can be decomposed into pieces which admit a geometric structure. We describe the details in the following.<br />
<br />
=== Prime decomposition === <br />
<wikitex>;<br />
{{beginthm|Definition}}<br />
A manifold $M$ is called prime, if it can't be written as a non-trivial connected sum, i.e. $M=M_1 \# M_2$ implies $M_1 = S^3$ or $M_2 = S^3$.<br />
A manifold $M$ is called irreducible if every embedded $S^2$ bounds a ball, i.e. the embedding extends to an embedding of $D^3$<br />
{{endthm}}<br />
Irreducibility is only slightly stronger than being prime. A orientable prime 3-manifold is either $S^2 \times S^1$ or every embedded 2-sphere bounds a ball.<br />
<br />
\begin{theorem}[Kneser]<br />
Every orientable, compact 3-manifold $M$ has a decomposition $M=P_1 \# \ldots \# P_n$ into prime manifolds $P_i$ unique up to ordering and $S^3$ summands.<br />
\end{theorem} <br />
<br />
A orientable prime 3-manifold is either $S^2 \times S^1$ or every embedded 2-sphere bounds a ball, in which case the manifold is called irreducible.<br />
<br />
Van Kampen's theorem tells you, that $\pi_1(M \# N)=\pi_1(M)*\pi_1(N)$. Hence any 3-manifold, whose fundamental group cannot be written as a free product of two nontrivial subgroups, can only be written as the connected sum of another 3-manifold with a simply connected 3-manifold. By the Poincaré conjecture a simply connected 3-manifold is already homeomorphic to $S^3$. Hence each such manifold is prime.<br />
<br />
Prime 3-manifolds can be distinguished by their fundamental groups into the following 3 types:<br />
</wikitex><br />
==== Type I: finite fundamental group==== <br />
<wikitex>;<br />
The universal cover $\tilde{M}$ is a simply-connected 3-manifold. As the fundamental group already determines the homology of a oriented, closed compact 3-manifold, it has to be a homology sphere. Using the Hurewicz-theorem, its fundamental class is represented by a degree 1 map $S^3 \rightarrow \tilde{M}$. This map induces isomorphisms on the homology and on the fundamental group. Hence it is a weak homotopy equivalence, and hence a homotopy equivalence by Whitehead's theorem (ref?).<br />
Hence every prime $3$-manifold with finite fundamental group arises as the quotient of a homotopy sphere by a free action of a finite group.<br />
With the use of the Poincaré conjecture every homotopy 3-sphere is homeomorphic to $S^3$ and we can write $M=S^3/\Gamma$. If $\Gamma$ is cyclic $M$ is known as lens space (ref). <br />
</wikitex><br />
==== Type II: infinite cyclic fundamental group ====<br />
<wikitex>;<br />
$S^1\times S^2$ is the only orientable closed prime 3-manifold of this type. Futhermore it is the only not irreducible prime manifold. (TODO: proof/ref)<br />
</wikitex><br />
<br />
==== Type III: infinite non-cyclic fundamental group ====<br />
<wikitex>;<br />
Such a manifold $M$ is always aspherical (TODO ref). The sphere theorem states, that every map $S^2\rightarrow M$ is homotopic to an embedding; and - as $M$ is irreducible - it is nullhomotopic. Hence $\pi_2(M)=0$. Consider the universal covering $\tilde{M}$ of $M$. Its first homology vanishes as it is simply connected. The long exact sequence of homotopy groups of the fibration $pi_1(M)\rightarrow \tilde{M}\rightarrow M$ gives a isomorphism $pi_2(M)\cong \pi_2(\tilde{M})$. Hence by Hurewicz' theorem $H_2(\tilde{M})=0$. Furthermore $H_3(\tilde{M})=0$, as $M$ is noncompact. Applying Hurewicz theorem again we get that all homotopy groups of $\tilde{M}$ vanish and hence by Whitehead's theorem $\tilde{M}$ is contractible. This means that $M$ is apherical. <br />
Hence the homotopy type of a prime 3-manifold with infinite non-cyclic fundamental group is uniquely determined by its fundamental group.<br />
Furthermore not every group can occur as a fundamental group of a prime 3-manifold. The equivariant cellular chain complex of $\tilde{M}$ is a projective resolution of the trivial $\Zz[\pi_1(M)]$-module $\Zz$. Hence ....<br />
For any subgroup $F\le \pi_1(M)$ the space $\tilde{M}/F$ is a finite-dimensional model for $K(F,1)$. For example a finite group cannot have such a model (by group homology ref) and hence $\pi_1(M)$ must be torsionfree. Furthermore it is a Poincaré duality group (link).<br />
</wikitex><br />
<br />
=== Torus decomposition ===<br />
<wikitex>;<br />
According to the previous section it remains to classify irreducible prime 3-manifolds. <br />
After cutting along spheres which don't bound balls as far as possible the next canonical step is to consider incompressible tori which are disjoint from the boundary. <br />
<br />
\begin{theorem}[Jacob-Shalen, Johannson]<br />
If $M$ is an irreducible compact orientable manifold, then there is a collection of disjoint incompressible tori $T_1, \ldots ,T_n$ in $M$ such that splitting $M$ along the union of these tori produces manifolds $M_i$ which are either [[Wikipedia:Seifert_fiber_space|Seifert-fibered]] or atoroidal, i.e. every incompressible torus in $M_i$ is isotopic to a torus component of $\partial M_i$. Furthermore, a minimal such collection of tori $T_j$ is unique up to isotopy in $M$.<br />
\end{theorem}<br />
<br />
Thurston's geometrization conjectures states that all the pieces we get by this JSJ-decomposition admit one of eight possible geometric structures:<br />
There is a list of eight simply connected Riemannian manifolds - the so called model geometries. A geometric structure on $M$ is the choice of a Riemannian metric on $M$, with the property that its universal covering $ \tilde{M}$ equipped with the pull-back metric is isometric to one of the eight model geometries. It might a priori be easier to classify all cocompact actions on the several model geometries.<br />
<br />
The Seifert-fibered pieces are well understood since the work of Seifert in the 30s (TODO: mention classification theorem). The atoroidal pieces are described by the following Hyperbolization theorem which was stated by Thurston (ref) and proven by Perelman.<br />
\begin{thm}<br />
Every irreducible atoroidal closed 3-manifold that is not Seifert-fibred is hyperbolic.<br />
\end{thm}<br />
</wikitex><br />
<br />
=== Dehn surgery ===<br />
<wikitex><br />
Dehn surgery is a way of constructing closed, oriented 3-manifolds. Given a [[Wikipedia:Link_%28knot_theory%29 | link]] in a $3$-manifold $N$<br />
$$L: \coprod_{i=1}^n S^1\rightarrow N,$$<br />
and a choice of a tubular neighborhood of $L$ <br />
$$L': \coprod_{i=1}^n S^1\times D^2\rightarrow N\mbox{ with }L'(x,0)=L(x)$$.<br />
(This choice essentially is the choice of a trivialization of the normal bundle; TODO find a correct formulation for this).<br />
This gives us a family of embedded, disjoint, full tori. The idea of Dehn surgery is to remove these Tori and glue them back in using a twist.\\<br />
Let us restrict to the case with only one solid torus $L':S^1\times D^2\rightarrow N$. <br />
Choose any self-homeomorphism $f$ of the torus $S^1\times S^1$. The result of the Dehn surgery at $L$ with the twist $f$ is defined as <br />
$$N_{f,L'}:=N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N\setminus L'(S^1\times \mathring{D}^2) \amalg S^1\times D^2/\sim,$$<br />
where the equivalence relation identifies for $(x,y)\in S^1\times S^1$ the points $L'(x,y)$ in the left component and $f(x,y)$ in the right component.<br />
If $f$ is the coordinate flipping, Dehn surgery is nothing but usual codimension $2$ surgery.<br />
<br />
\begin{lemma}<br />
Suppose $f,f'\in \Homeo(T^2)$ are isotopic and let $L':S^1\times D^2 \rightarrow N$ be any embedding of the full Torus in a $3$-Manifold $N$.Then $N_{f,L'}$ and $N_{f',L'}$ are homeomorphic.<br />
\end{lemma}<br />
\begin{proof}<br />
Let $j:T^2\times [0;1] \rightarrow T^2$ be an isotopy from $f$ to $f'$. This gives a homeomorphism:<br />
$$ J:T^2\times [0;1]\rightarrow T^2\times [0;1] \qquad (x,y,t)\mapsto (j(x,y,t),t). $$<br />
(TODO: is its inverse $(x,y,t)\mapsto (j(-,-,t)^{-1}(x,y),t)$ continuous ?).<br />
The idea is to grab some additional space, where one can use the map $J$. TODO<br />
\end{proof}<br />
TODO formulate a lemma, that M_{f,L'} also only depends on the isotopy class of $L'$ (which is hopefully true).<br />
Hence, we have to classify all self-homeomorphisms of $T^2$ up to isotopy.<br />
\begin{lemma}<br />
Every self-homeomorphism of $T^2$ is isotopic to exactly one homeomorphism of the shape <br />
$$f_A:\Rr^2/\Zz^2 \rightarrow \Rr^2 /\Zz^2 \qquad \left(\begin{array}{c}x\\y\end{array}\right)\mapsto A\cdot\left(\begin{array}{c}x\\y\end{array}\right),$$<br />
where $A\in GL_2(\Zz)$ (reference of proof). <br />
\end{lemma}<br />
\begin{proof} Since the torus is a $K\left(\Zz^2,1\right)$-space, we have that $\pi_0 Map\left(T^2,T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is an isomorphism. Homotopic surface homeomorphisms are isotopic (Reference?). Thus the restriction $\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is injective. Moreover, each $A\in Hom\left(\Zz^2,\Zz^2\right)$ is realised by $f_A$, therefore $\pi_0Homeo\left(T^2\right)\rightarrow Hom\left(\Zz^2,\Zz^2\right)$ is also surjective. <br />
TODO find a reference in ANY source about the mapping class group<br />
\end{proof}<br />
The next lemma tells us, that composition of self-homeomorphisms corresponds to two successive Dehn surgeries.<br />
\begin{lemma}<br />
Let $f,g\in \Homeo(T^2)$ be given and let $L':S^1\times D^2\rightarrow N$ is an embedding of the full torus in a 3-manifold. Then we have map <br />
$$L'':S^1 \times D^2 \rightarrow N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N_{f,L'}$$<br />
given by the map $S^1\times D^2\rightarrow S^1 \times D^2 \quad (x,y)(x,y/2)$ postcomposed with the canonical inclusion in the second coordinate.<br />
Then $(N_{f,L'})_{g,L''} \cong N_{f\circ g,L'}$. TODO right order of composition ? We will see in the proof.<br />
\end{lemma}<br />
\beg{proof}<br />
\end{proof}<br />
<br />
We have to find out, which self-homeomorphisms of the torus don't change the homeomorphism type of the manifold.<br />
\begin{lemma} Consider a matrix of the form $\left( \begin{array}{cc} 1 & 0 \\ k & 1 \end{array}\right)$ and let $L':S^1\times D^2 \rightarrow N$ be any eembedding. Then $N_{f_A,L'}\cong N$.<br />
\end{lemma}<br />
TODO are there any orientation reversing homeos, that also extend ? Think so. Also add them here.<br />
\begin{proof}<br />
The homeomorphism $f_A \in \Homeo(T^2)$ extends to a homeomorphism of $\bar{f_A}\in \Homeo(S^1\times D^2)$:<br />
$$\bar{f_A}(x,y):=(x,x^ky), $$<br />
where $x\in S^1 = \{z\in \Cc| |z|=1\}, y\in D^2=\{y\in\Cc||y|\le 1\}$.<br />
Using this homeomorphism one can define a homeomorphism from $N_f$ to $N$:<br />
$$N = N\setminus L(S^1\times \mathring{D}^2)\cup_1 S^1\times D^2 \rightarrow N_{f_A}=N\setminus L(S^1 \times \mathring{D}^2)\cup_{f_A} S^1\times D^2$$<br />
given by the identity on the left component and $\bar{f_A}$ on the right component.<br />
\end{proof}<br />
<br />
Together with (link to comment about composition), this tells us, that $N_{f_A}$ really only depends on the coset $A\cdot \left(\begin{array}{cc}1&*\\0&1\end{array}\right)$ (TODO check right or left coset). This coset is uniquely determined by the image $(p,q)$ of $(1,0)$ with $p$ and $q$ coprime.<br />
<br />
The ratio $p/q$ is called the surgery coefficient. (TODO what is the quotient good for ?)<++><br />
<br />
<br />
TODO does the result give different manifolds.<br />
<br />
TODO does the result only depend on the isotopy class of the link. <br />
<br />
Every compact (oriented /able, neccesary ?) 3-manifold might be obtained from $S^3$ by a Dehn surgery along a link (TODO ref). Of course this does not satisfy to classify 3-manifolds without having a good classification of links in $S^3$. <br />
<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
\cite{Scott1983}, \cite{Thurston1997}, \cite{Hatcher2000}, \cite{Hempel1976}<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/3-manifolds3-manifolds2010-06-18T17:10:19Z<p>Kuessner: /* Dehn surgery */</p>
<hr />
<div>{{Stub}}<br />
<br />
== Introduction ==<br />
<wikitex>;<br />
In the 3-dimensional setting there is no distinction between smooth, PL and topological manifolds neccesary; the categories of smooth, PL and topological manifolds are equivalent (TODO ref). A lot of techniques have been developed in the last century to study 3-manifolds but most of them are very special and don't generalise to higher dimensions. One key idea is to decompose manifolds along incompressible surfaces into smaller pieces, to which certain geometric models apply. <br />
A great progress was made in with the proof of the Poincaré conjecture and Thurton's geometrization conjecture by Perelman in 2003.<br />
[[Image:Phomsphere.jpg|thumb|150px| The universal cover of the famous [[Poincaré homology sphere]] is $S^3$ - here a view of the induced tesselation]]<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<br />
<wikitex>;<br />
Basic examples are $\mathbb{R}^3, S^3, S^1 \times S$ with $S$ any surface. <br />
Important types of 3-manifolds are [[Wikipedia:Haken_manifold|Haken-Manifolds]], [[Wikipedia:Seifert_fibre_spaces|Seifert-Manifolds]], [[Wikipedia:Lens_spaces|3-dimensional lens spaces]], [[Wikipedia:Torus_bundle|Torus-bundles and Torus semi-bundles]].<br />
<br />
There are two topological processes to join 3-manifolds to get a new one.<br />
The first is the connected sum of two manifolds $M_1$ and $M_2$. Choose embeddings $f_1:D^3\rightarrow M_1$ and $f_2:D^3\rightarrow M_2$, remove the interior of $f_1(D^3)$ and $f_2(D^3)$ and glue $M_1$ and $M_2$ together along the boundaries $f_1(S^3)$ and $f_2(S^3)$.<br />
The second uses incompressible surfaces. Let $M$ be manifold and $S\subset M$ a surface. $S$ is incompressible, if there is no disk $D$ in $M$ with $D\cap S=\partial D$.<br />
The torus sum is the process which glues incompressible tori boundary components together.<br />
<br />
(TODO What is incompressibility needed? / What is is good for/ What happen if one takes a compressible surface ?)<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
In the 3-dimensional world the fundamental group is a powerful invariant to distiguish manifolds.<br />
It determines already all homology groups: <br />
* $H_1(M)$ = abelization of $\pi_1(M)$. <br />
* $H_2(M) = H^1(M) = H_1(M)/$torsion<br />
* $H_3(M) = \Zz$<br />
* $H_n(M) = 0$ for $n > 3$ <br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
By reversing the process of connected and torus sum every 3-manifold can be decomposed into pieces which admit a geometric structure. We describe the details in the following.<br />
<br />
=== Prime decomposition === <br />
<wikitex>;<br />
{{beginthm|Definition}}<br />
A manifold $M$ is called prime, if it can't be written as a non-trivial connected sum, i.e. $M=M_1 \# M_2$ implies $M_1 = S^3$ or $M_2 = S^3$.<br />
A manifold $M$ is called irreducible if every embedded $S^2$ bounds a ball, i.e. the embedding extends to an embedding of $D^3$<br />
{{endthm}}<br />
Irreducibility is only slightly stronger than being prime. A orientable prime 3-manifold is either $S^2 \times S^1$ or every embedded 2-sphere bounds a ball.<br />
<br />
\begin{theorem}[Kneser]<br />
Every orientable, compact 3-manifold $M$ has a decomposition $M=P_1 \# \ldots \# P_n$ into prime manifolds $P_i$ unique up to ordering and $S^3$ summands.<br />
\end{theorem} <br />
<br />
A orientable prime 3-manifold is either $S^2 \times S^1$ or every embedded 2-sphere bounds a ball, in which case the manifold is called irreducible.<br />
<br />
Van Kampen's theorem tells you, that $\pi_1(M \# N)=\pi_1(M)*\pi_1(N)$. Hence any 3-manifold, whose fundamental group cannot be written as a free product of two nontrivial subgroups, can only be written as the connected sum of another 3-manifold with a simply connected 3-manifold. By the Poincaré conjecture a simply connected 3-manifold is already homeomorphic to $S^3$. Hence each such manifold is prime.<br />
<br />
Prime 3-manifolds can be distinguished by their fundamental groups into the following 3 types:<br />
</wikitex><br />
==== Type I: finite fundamental group==== <br />
<wikitex>;<br />
The universal cover $\tilde{M}$ is a simply-connected 3-manifold. As the fundamental group already determines the homology of a oriented, closed compact 3-manifold, it has to be a homology sphere. Using the Hurewicz-theorem, its fundamental class is represented by a degree 1 map $S^3 \rightarrow \tilde{M}$. This map induces isomorphisms on the homology and on the fundamental group. Hence it is a weak homotopy equivalence, and hence a homotopy equivalence by Whitehead's theorem (ref?).<br />
Hence every prime $3$-manifold with finite fundamental group arises as the quotient of a homotopy sphere by a free action of a finite group.<br />
With the use of the Poincaré conjecture every homotopy 3-sphere is homeomorphic to $S^3$ and we can write $M=S^3/\Gamma$. If $\Gamma$ is cyclic $M$ is known as lens space (ref). <br />
</wikitex><br />
==== Type II: infinite cyclic fundamental group ====<br />
<wikitex>;<br />
$S^1\times S^2$ is the only orientable closed prime 3-manifold of this type. Futhermore it is the only not irreducible prime manifold. (TODO: proof/ref)<br />
</wikitex><br />
<br />
==== Type III: infinite non-cyclic fundamental group ====<br />
<wikitex>;<br />
Such a manifold $M$ is always aspherical (TODO ref). The sphere theorem states, that every map $S^2\rightarrow M$ is homotopic to an embedding; and - as $M$ is irreducible - it is nullhomotopic. Hence $\pi_2(M)=0$. Consider the universal covering $\tilde{M}$ of $M$. Its first homology vanishes as it is simply connected. The long exact sequence of homotopy groups of the fibration $pi_1(M)\rightarrow \tilde{M}\rightarrow M$ gives a isomorphism $pi_2(M)\cong \pi_2(\tilde{M})$. Hence by Hurewicz' theorem $H_2(\tilde{M})=0$. Furthermore $H_3(\tilde{M})=0$, as $M$ is noncompact. Applying Hurewicz theorem again we get that all homotopy groups of $\tilde{M}$ vanish and hence by Whitehead's theorem $\tilde{M}$ is contractible. This means that $M$ is apherical. <br />
Hence the homotopy type of a prime 3-manifold with infinite non-cyclic fundamental group is uniquely determined by its fundamental group.<br />
Furthermore not every group can occur as a fundamental group of a prime 3-manifold. The equivariant cellular chain complex of $\tilde{M}$ is a projective resolution of the trivial $\Zz[\pi_1(M)]$-module $\Zz$. Hence ....<br />
For any subgroup $F\le \pi_1(M)$ the space $\tilde{M}/F$ is a finite-dimensional model for $K(F,1)$. For example a finite group cannot have such a model (by group homology ref) and hence $\pi_1(M)$ must be torsionfree. Furthermore it is a Poincaré duality group (link).<br />
</wikitex><br />
<br />
=== Torus decomposition ===<br />
<wikitex>;<br />
According to the previous section it remains to classify irreducible prime 3-manifolds. <br />
After cutting along spheres which don't bound balls as far as possible the next canonical step is to consider incompressible tori which are disjoint from the boundary. <br />
<br />
\begin{theorem}[Jacob-Shalen, Johannson]<br />
If $M$ is an irreducible compact orientable manifold, then there is a collection of disjoint incompressible tori $T_1, \ldots ,T_n$ in $M$ such that splitting $M$ along the union of these tori produces manifolds $M_i$ which are either [[Wikipedia:Seifert_fiber_space|Seifert-fibered]] or atoroidal, i.e. every incompressible torus in $M_i$ is isotopic to a torus component of $\partial M_i$. Furthermore, a minimal such collection of tori $T_j$ is unique up to isotopy in $M$.<br />
\end{theorem}<br />
<br />
Thurston's geometrization conjectures states that all the pieces we get by this JSJ-decomposition admit one of eight possible geometric structures:<br />
There is a list of eight simply connected Riemannian manifolds - the so called model geometries. A geometric structure on $M$ is the choice of a Riemannian metric on $M$, with the property that its universal covering $ \tilde{M}$ equipped with the pull-back metric is isometric to one of the eight model geometries. It might a priori be easier to classify all cocompact actions on the several model geometries.<br />
<br />
The Seifert-fibered pieces are well understood since the work of Seifert in the 30s (TODO: mention classification theorem). The atoroidal pieces are described by the following Hyperbolization theorem which was stated by Thurston (ref) and proven by Perelman.<br />
\begin{thm}<br />
Every irreducible atoroidal closed 3-manifold that is not Seifert-fibred is hyperbolic.<br />
\end{thm}<br />
</wikitex><br />
<br />
=== Dehn surgery ===<br />
<wikitex><br />
Dehn surgery is a way of constructing closed, oriented 3-manifolds. Given a [[Wikipedia:Link_%28knot_theory%29 | link]] in a $3$-manifold $N$<br />
$$L: \coprod_{i=1}^n S^1\rightarrow N,$$<br />
and a choice of a tubular neighborhood of $L$ <br />
$$L': \coprod_{i=1}^n S^1\times D^2\rightarrow N\mbox{ with }L'(x,0)=L(x)$$.<br />
(This choice essentially is the choice of a trivialization of the normal bundle; TODO find a correct formulation for this).<br />
This gives us a family of embedded, disjoint, full tori. The idea of Dehn surgery is to remove these Tori and glue them back in using a twist.\\<br />
Let us restrict to the case with only one solid torus $L':S^1\times D^2\rightarrow N$. <br />
Choose any self-homeomorphism $f$ of the torus $S^1\times S^1$. The result of the Dehn surgery at $L$ with the twist $f$ is defined as <br />
$$N_{f,L'}:=N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N\setminus L'(S^1\times \mathring{D}^2) \amalg S^1\times D^2/\sim,$$<br />
where the equivalence relation identifies for $(x,y)\in S^1\times S^1$ the points $L'(x,y)$ in the left component and $f(x,y)$ in the right component.<br />
If $f$ is the coordinate flipping, Dehn surgery is nothing but usual codimension $2$ surgery.<br />
<br />
\begin{lemma}<br />
Suppose $f,f'\in \Homeo(T^2)$ are isotopic and let $L':S^1\times D^2 \rightarrow N$ be any embedding of the full Torus in a $3$-Manifold $N$.Then $N_{f,L'}$ and $N_{f',L'}$ are homeomorphic.<br />
\end{lemma}<br />
\begin{proof}<br />
Let $j:T^2\times [0;1] \rightarrow T^2$ be an isotopy from $f$ to $f'$. This gives a homeomorphism:<br />
$$ J:T^2\times [0;1]\rightarrow T^2\times [0;1] \qquad (x,y,t)\mapsto (j(x,y,t),t). $$<br />
(TODO: is its inverse $(x,y,t)\mapsto (j(-,-,t)^{-1}(x,y),t)$ continuous ?).<br />
The idea is to grab some additional space, where one can use the map $J$. TODO<br />
\end{proof}<br />
TODO formulate a lemma, that M_{f,L'} also only depends on the isotopy class of $L'$ (which is hopefully true).<br />
Hence, we have to classify all self-homeomorphisms of $T^2$ up to isotopy.<br />
\begin{lemma}<br />
Every self-homeomorphism of $T^2$ is isotopic to exactly one homeomorphism of the shape <br />
$$f_A:\Rr^2/\Zz^2 \rightarrow \Rr^2 /\Zz^2 \qquad \left(\begin{array}{c}x\\y\end{array}\right)\mapsto A\cdot\left(\begin{array}{c}x\\y\end{array}\right),$$<br />
where $A\in GL_2(\Zz)$ (reference of proof). <br />
\end{lemma}<br />
\begin{proof}<br />
TODO find a reference in ANY source about the mapping class group<br />
\end{proof}<br />
The next lemma tells us, that composition of self-homeomorphisms corresponds to two successive Dehn surgeries.<br />
\begin{lemma}<br />
Let $f,g\in \Homeo(T^2)$ be given and let $L':S^1\times D^2\rightarrow N$ is an embedding of the full torus in a 3-manifold. Then we have map <br />
$$L'':S^1 \times D^2 \rightarrow N\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2=N_{f,L'}$$<br />
given by the map $S^1\times D^2\rightarrow S^1 \times D^2 \quad (x,y)(x,y/2)$ postcomposed with the canonical inclusion in the second coordinate.<br />
Then $(N_{f,L'})_{g,L''} \cong N_{f\circ g,L'}$. TODO right order of composition ? We will see in the proof.<br />
\end{lemma}<br />
\beg{proof}<br />
\end{proof}<br />
<br />
We have to find out, which self-homeomorphisms of the torus don't change the homeomorphism type of the manifold.<br />
\begin{lemma} Consider a matrix of the form $\left( \begin{array}{cc} 1 & 0 \\ k & 1 \end{array}\right)$ and let $L':S^1\times D^2 \rightarrow N$ be any eembedding. Then $N_{f_A,L'}\cong N$.<br />
\end{lemma}<br />
TODO are there any orientation reversing homeos, that also extend ? Think so. Also add them here.<br />
\begin{proof}<br />
The homeomorphism $f_A \in \Homeo(T^2)$ extends to a homeomorphism of $\bar{f_A}\in \Homeo(S^1\times D^2)$:<br />
$$\bar{f_A}(x,y):=(x,x^ky), $$<br />
where $x\in S^1 = \{z\in \Cc| |z|=1\}, y\in D^2=\{y\in\Cc||y|\le 1\}$.<br />
Using this homeomorphism one can define a homeomorphism from $N_f$ to $N$:<br />
$$N = N\setminus L(S^1\times \mathring{D}^2)\cup_1 S^1\times D^2 \rightarrow N_{f_A}=N\setminus L(S^1 \times \mathring{D}^2)\cup_{f_A} S^1\times D^2$$<br />
given by the identity on the left component and $\bar{f_A}$ on the right component.<br />
\end{proof}<br />
<br />
Together with (link to comment about composition), this tells us, that $N_{f_A}$ really only depends on the coset $A\cdot \left(\begin{array}{cc}1&*\\0&1\end{array}\right)$ (TODO check right or left coset). This coset is uniquely determined by the image $(p,q)$ of $(1,0)$ with $p$ and $q$ coprime.<br />
<br />
The ratio $p/q$ is called the surgery coefficient. (TODO what is the quotient good for ?)<++><br />
<br />
<br />
TODO does the result give different manifolds.<br />
<br />
TODO does the result only depend on the isotopy class of the link. <br />
<br />
Every compact (oriented /able, neccesary ?) 3-manifold might be obtained from $S^3$ by a Dehn surgery along a link (TODO ref). Of course this does not satisfy to classify 3-manifolds without having a good classification of links in $S^3$. <br />
<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
\cite{Scott1983}, \cite{Thurston1997}, \cite{Hatcher2000}, \cite{Hempel1976}<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationsFoliations2010-06-14T11:17:37Z<p>Kuessner: </p>
<hr />
<div><!-- COMMENT: <br />
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To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
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--><br />
{{Stub}}<br />
<br />
== Introduction ==<br />
<wikitex>;<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
=== Bundles ===<br />
<br />
<wikitex>;<br />
<br />
The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)<br />
<br />
A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth) manifold $M$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)<br />
<br />
Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a diffeomorphism $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:<br />
$$\begin{xy}<br />
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\<br />
U\ar[r]^{id}&U}<br />
\end{xy}$$<br />
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.<br />
</wikitex><br />
<br />
=== Suspensions ===<br />
<wikitex>;<br />
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=<br />
\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\mbox{\ for\ }f\in F,$$<br />
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.<br />
<br />
In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.<br />
<br />
The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. <br />
If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. <br />
</wikitex><br />
<br />
=== Submersions ===<br />
<wikitex>;<br />
Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. <br />
[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]<br />
An example of a submersion, which is not a fiber bundle, is given by<br />
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$<br />
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$<br />
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.<br />
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$<br />
for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the M&ouml;bius band). Their leaf spaces are not Hausdorff.<br />
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]<br />
</wikitex><br />
<br />
=== Reeb foliations ===<br />
<wikitex>;<br />
Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by<br />
$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$<br />
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ <br />
for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.<br />
<br />
</wikitex><br />
<br />
=== Taut foliations ===<br />
<wikitex>;<br />
A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. <br />
<br />
{{beginthm|Theorem|(Rummler, Sullivan)}}<br />
The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$:<br />
<br />
a) $\mathcal{F}$ is taut;<br />
<br />
b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$;<br />
<br />
c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}} <br />
<br />
<br />
</wikitex><br />
<br />
=== Constructing new foliations from old ones ===<br />
<br />
==== Pullbacks ====<br />
<wikitex>;<br />
{{beginthm|Theorem |}} If $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ and $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$, then $N$ is foliated by connected components of $f^{-1}\left(L\right)$ as $L$ ranges over the leaves of $\mathcal{F}$. {{endthm}}<br />
{{cite|Candel&Conlon2000}}, Theorem 3.2.2<br />
</wikitex><br />
<br />
==== Glueing ====<br />
<wikitex>;<br />
Let $\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$ be $n$-manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$.<br />
If either both foliations are tangent or both foliations are transverse to the boundaries of $M_1$ and $M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$. This is called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$.<br />
</wikitex><br />
<br />
==== Turbulization ====<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$. <br />
<br />
Define a foliation ${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by<br />
$$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$<br />
where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$<br />
The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhood of the boundary of $N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$ of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<br />
=== Holonomy ===<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation and $L$ a leaf. For a path $\gamma:\left[0,1\right]\rightarrow L$ contained in the intersection of the leaf $L$ with<br />
a foliation chart $U$, and two transversals $\tau_0,\tau_1$ to $\gamma$ at the endpoints, the product structure of the foliation chart determines a homeomorphism $$h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$$<br />
If $\gamma$ is covered by foliation charts $U_0,\ldots,U_k$, then one obtains a sequence of homeomorphisms $$h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$$<br />
The composition yields a well-defined map $h$ from the germ of $\tau_0$ at $\gamma\left(0\right)$<br />
to the germ of $\tau_{k+1}$ at $\gamma\left(1\right)$, the so-called holonomy transport. The holonomy transport only depends on the relative homotopy class of $\gamma$. <br />
{{beginthm|Lemma}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation, $L$ a leaf, $x\in L$ and $\tau$ a transversal at $x$. Holonomy transport defines a homomorphism $$H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)$$ to the group of germs of homeomorphisms of $\tau$.<br />
{{endthm}}<br />
<br />
{{beginthm|Corollary|(Reeb)}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension one foliation of a 3-manifold such that some leaf $L$ is homeomorphic to $S^2$. Then $M=S^2\times S^1$ and $\mathcal{F}$ is the product foliation by spheres.<br />
{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.5<br />
<br />
<br />
</wikitex><br />
=== Godbillon-Vey invariant ===<br />
<wikitex>;<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation of a manifold $M$, then its tangential plane field $E$ is defined by a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ and $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. The Godbillon-Vey invariant of $\mathcal{F}$ is defined as<br />
$$gv\left({\mathcal{F}}\right):=\left[\eta\wedge\left(d\eta\right)^q\right]\in H^{2q+1}_{dR}\left(M\right).$$<br />
<br />
The Godbillion-Vey invariant is related to resilience of leaves. A leaf is said to be resilient if it is not properly embedded and its holonomy is not trivial.<br />
<br />
{{beginthm|Theorem|(Duminy)}}<br />
If $\left(M,{\mathcal{F}}\right)$ is a foliation of codimension one and no leaf is resilient, then $gv\left({\mathcal{F}}\right)=0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== Classification ==<br />
<br />
=== Codimension one foliations ===<br />
<br />
==== Existence ====<br />
<wikitex>;<br />
{{beginthm|Theorem}}<br />
A closed smooth manifold $M^n$ has a smooth codimension one foliation if and only if $\chi(M^n)=0$, where $\chi$ denotes the Euler characteristic. <br />
<br />
If $\chi(M^n)=0$, then every $(n-1)$-plane field $\tau^{n-1}$ on $M^n$ is homotopic to the tangent plane field of a smooth codimension one foliation.{{endthm}}<br />
<br />
{{cite|Thurston1976}}<br />
</wikitex><br />
<br />
==== Foliations of surfaces ====<br />
<wikitex>;<br />
If $\left({\mathbb R}^2,{\mathcal{F}}\right)$ is a codimension one foliation of the plane ${\mathbb R}^2$, then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold $L$. This provides a 1-1-correspondence between foliations of ${\mathbb R}^2$ and simply connected 1-manifolds. <br />
<br />
Codimension one foliations on compact surfaces $S$ exist only if $\chi\left(S\right)=0$, that is on the Torus, the Klein bottle, the annulus and the Möbius band.<br />
<br />
A foliation $\left(S,{\mathcal{F}}\right)$ is said to contain a Reeb component resp. a non-orientable Reeb component if the restriction of ${\mathcal{F}}$ to some subsurface $S^\prime\subset S$ is a Reeb foliation resp. a non-orientable Reeb foliation. (This implies that $S^\prime$ is an annulus resp. a Möbius band.)<br />
{{beginthm|Theorem}} <br />
<br />
a) Let $\left(S,{\mathcal{F}}\right)$ be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:<br />
<br />
(1) $\mathcal{F}$ is the suspension of a homeomorphism $f:S^1\rightarrow S^1$ or<br />
<br />
(2) $\mathcal{F}$ contains a Reeb component (orientable or not).<br />
<br />
b) Every foliation of the annulus $S^1\times I$ tangent to the boundary is obtained by glueing together a finite number of Reeb components and a finite number of suspensions<br />
<br />
c) Every foliation of the Möbius band tangent to the boundary is one of the following three possibly glued together with a foliation on $S^1\times I$:<br />
<br />
(1) the non-orientable Reeb component<br />
<br />
(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution<br />
<br />
(3) the suspension of an orientation-reversing homeomorpism $f:I\rightarrow I$.{{endthm}}<br />
{{cite|Hector&Hirsch1981}}, Theorem 4.2.15 and Proposition 4.3.2<br />
</wikitex><br />
<br />
==== Foliations of 3-manifolds ====<br />
<wikitex>;<br />
{{beginthm|Theorem|(Novikov)}} If a 3-manifold $M$ admits a foliation $\mathcal{F}$ without Reeb components, then $\pi_2\left(M\right)=0$, every leaf of $\mathcal{F}$ is incompressible, and every transverse loop is essential in $\pi_1\left(M\right)$. {{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.37<br />
<br />
A taut foliation has no Reeb component. If $M$ is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.<br />
<br />
{{beginthm|Theorem|(Palmeira)}} If $\mathcal{F}$ is a taut foliation of a 3-manifold $M$ not finitely covered by $S^2\times S^1$, then the universal covering $\widetilde{M}$ is homeomorphic to ${\mathbb R}^3$ and the pull-back foliation $\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$ is homeomorphic to a product foliation $\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$, where $\mathcal{G}$ is a foliation of ${\mathbb R}^2$ by lines.{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.38<br />
<br />
In particular, a taut foliation of a 3-manifold $M$ yields an action of $\pi_1M$ on a (possibly non-Hausdorff) simply connected 1-manifold $L$, the space of leaves of ${\mathcal{G}}$.<br />
<br />
{{beginthm|Theorem|(Gabai)}} Let $M$ be a closed, irreducible 3-manifold.<br />
<br />
a) If $H_2\left(M;{\mathbb R}\right)\not =0$, then $M$ admits a taut foliation.<br />
<br />
b) If $S$ is a surface which minimizes the Thurston norm in its homology class $\left[S\right]\in H_2\left(M;{\mathbb R}\right)$, then $M$ admits a taut foliaton for which $S$ is a leaf.{{endthm}}<br />
<br />
=== Codimension two foliations ===<br />
<br />
==== $S^1$-foliations of 3-manifolds ====<br />
<br />
{{beginthm|Theorem|(Epstein)}} Every foliation of a compact 3-manifold by circles is a Seifert fibration.{{endthm}}<br />
<br />
{{beginthm|Example}}<br />
a) For every rational number $\frac{p}{q}\not=0$ there exists a foliaton of $S^3=\left\{\left(z,w\right)\in{\mathbb C}^2: \mid z\mid^2+\mid w\mid^2=1\right\}$ by circles such that restriction to the standard embedded torus $\left\{\left(z,w\right)\in S^3: \mid z\mid=\mid w\mid=1\right\}$ is the suspension foliation of $R_{\frac{p}{q}2\pi}$. <br />
b) The complement of a knot $K\subset S^3$ admits a foliation by circles if and only if $K$ is a torus knot.{{endthm}}<br />
<br />
{{beginthm|Theorem|(Vogt)}} If a 3-manifold $M$ admits a foliation by circles, then any 3-manifold obtained by removing finitely many points from $M$ admits a (not necessarily smooth) foliation by circles.{{endthm}}<br />
<br />
{{beginthm|Corollary}} ${\mathbb R}^3$ admits a foliation by circles.{{endthm}}<br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
{{Stub}}</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationsFoliations2010-06-14T11:13:20Z<p>Kuessner: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
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{{Stub}}<br />
<br />
== Introduction ==<br />
<wikitex>;<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
=== Bundles ===<br />
<br />
<wikitex>;<br />
<br />
The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)<br />
<br />
A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth) manifold $M$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)<br />
<br />
Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a diffeomorphism $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:<br />
$$\begin{xy}<br />
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\<br />
U\ar[r]^{id}&U}<br />
\end{xy}$$<br />
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.<br />
</wikitex><br />
<br />
=== Suspensions ===<br />
<wikitex>;<br />
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=<br />
\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\mbox{\ for\ }f\in F,$$<br />
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.<br />
<br />
In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.<br />
<br />
The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. <br />
If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. <br />
</wikitex><br />
<br />
=== Submersions ===<br />
<wikitex>;<br />
Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. <br />
[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]<br />
An example of a submersion, which is not a fiber bundle, is given by<br />
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$<br />
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$<br />
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.<br />
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$<br />
for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the M&ouml;bius band). Their leaf spaces are not Hausdorff.<br />
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]<br />
</wikitex><br />
<br />
=== Reeb foliations ===<br />
<wikitex>;<br />
Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by<br />
$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$<br />
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ <br />
for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.<br />
<br />
</wikitex><br />
<br />
=== Taut foliations ===<br />
<wikitex>;<br />
A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. <br />
<br />
{{beginthm|Theorem|(Rummler, Sullivan)}}<br />
The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$:<br />
<br />
a) $\mathcal{F}$ is taut;<br />
<br />
b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$;<br />
<br />
c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}} <br />
<br />
<br />
</wikitex><br />
<br />
=== Constructing new foliations from old ones ===<br />
<br />
==== Pullbacks ====<br />
<wikitex>;<br />
{{beginthm|Theorem |}} If $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ and $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$, then $N$ is foliated by connected components of $f^{-1}\left(L\right)$ as $L$ ranges over the leaves of $\mathcal{F}$. {{endthm}}<br />
{{cite|Candel&Conlon2000}}, Theorem 3.2.2<br />
</wikitex><br />
<br />
==== Glueing ====<br />
<wikitex>;<br />
Let $\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$ be $n$-manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$.<br />
If either both foliations are tangent or both foliations are transverse to the boundaries of $M_1$ and $M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$. This is called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$.<br />
</wikitex><br />
<br />
==== Turbulization ====<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$. <br />
<br />
Define a foliation ${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by<br />
$$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$<br />
where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$<br />
The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhood of the boundary of $N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$ of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<br />
=== Holonomy ===<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation and $L$ a leaf. For a path $\gamma:\left[0,1\right]\rightarrow L$ contained in the intersection of the leaf $L$ with<br />
a foliation chart $U$, and two transversals $\tau_0,\tau_1$ to $\gamma$ at the endpoints, the product structure of the foliation chart determines a homeomorphism $$h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$$<br />
If $\gamma$ is covered by foliation charts $U_0,\ldots,U_k$, then one obtains a sequence of homeomorphisms $$h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$$<br />
The composition yields a well-defined map $h$ from the germ of $\tau_0$ at $\gamma\left(0\right)$<br />
to the germ of $\tau_{k+1}$ at $\gamma\left(1\right)$, the so-called holonomy transport. The holonomy transport only depends on the relative homotopy class of $\gamma$. <br />
{{beginthm|Lemma}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation, $L$ a leaf, $x\in L$ and $\tau$ a transversal at $x$. Holonomy transport defines a homomorphism $$H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)$$ to the group of germs of homeomorphisms of $\tau$.<br />
{{endthm}}<br />
<br />
{{beginthm|Corollary|(Reeb)}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension one foliation of a 3-manifold such that some leaf $L$ is homeomorphic to $S^2$. Then $M=S^2\times S^1$ and $\mathcal{F}$ is the product foliation by spheres.<br />
{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.5<br />
<br />
<br />
</wikitex><br />
=== Godbillon-Vey invariant ===<br />
<wikitex>;<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation of a manifold $M$, then its tangential plane field $E$ is defined by a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ and $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. The Godbillon-Vey invariant of $\mathcal{F}$ is defined as<br />
$$gv\left({\mathcal{F}}\right):=\left[\eta\wedge\left(d\eta\right)^q\right]\in H^{2q+1}_{dR}\left(M\right).$$<br />
<br />
The Godbillion-Vey invariant is related to resilience of leaves. A leaf is said to be resilient if it is not properly embedded and its holonomy is not trivial.<br />
<br />
{{beginthm|Theorem|(Duminy)}}<br />
If $\left(M,{\mathcal{F}}\right)$ is a foliation of codimension one and no leaf is resilient, then $gv\left({\mathcal{F}}\right)=0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== Classification ==<br />
<br />
=== Codimension one foliations ===<br />
<br />
==== Existence ====<br />
<wikitex>;<br />
{{beginthm|Theorem}}<br />
A closed smooth manifold $M^n$ has a smooth codimension one foliation if and only if $\chi(M^n)=0$, where $\chi$ denotes the Euler characteristic. <br />
<br />
If $\chi(M^n)=0$, then every $(n-1)$-plane field $\tau^{n-1}$ on $M^n$ is homotopic to the tangent plane field of a smooth codimension one foliation.{{endthm}}<br />
<br />
{{cite|Thurston1976}}<br />
</wikitex><br />
<br />
==== Foliations of surfaces ====<br />
<wikitex>;<br />
If $\left({\mathbb R}^2,{\mathcal{F}}\right)$ is a codimension one foliation of the plane ${\mathbb R}^2$, then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold $L$. This provides a 1-1-correspondence between foliations of ${\mathbb R}^2$ and simply connected 1-manifolds. <br />
<br />
Codimension one foliations on compact surfaces $S$ exist only if $\chi\left(S\right)=0$, that is on the Torus, the Klein bottle, the annulus and the Möbius band.<br />
<br />
A foliation $\left(S,{\mathcal{F}}\right)$ is said to contain a Reeb component resp. a non-orientable Reeb component if the restriction of ${\mathcal{F}}$ to some subsurface $S^\prime\subset S$ is a Reeb foliation resp. a non-orientable Reeb foliation. (This implies that $S^\prime$ is an annulus resp. a Möbius band.)<br />
{{beginthm|Theorem}} <br />
<br />
a) Let $\left(S,{\mathcal{F}}\right)$ be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:<br />
<br />
(1) $\mathcal{F}$ is the suspension of a homeomorphism $f:S^1\rightarrow S^1$ or<br />
<br />
(2) $\mathcal{F}$ contains a Reeb component (orientable or not).<br />
<br />
b) Every foliation of the annulus $S^1\times I$ tangent to the boundary is obtained by glueing together a finite number of Reeb components and a finite number of suspensions<br />
<br />
c) Every foliation of the Möbius band tangent to the boundary is one of the following three possibly glued together with a foliation on $S^1\times I$:<br />
<br />
(1) the non-orientable Reeb component<br />
<br />
(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution<br />
<br />
(3) the suspension of an orientation-reversing homeomorpism $f:I\rightarrow I$.{{endthm}}<br />
{{cite|Hector&Hirsch1981}}, Theorem 4.2.15 and Proposition 4.3.2<br />
</wikitex><br />
<br />
==== Foliations of 3-manifolds ====<br />
<wikitex>;<br />
{{beginthm|Theorem|(Novikov)}} If a 3-manifold $M$ admits a foliation $\mathcal{F}$ without Reeb components, then $\pi_2\left(M\right)=0$, every leaf of $\mathcal{F}$ is incompressible, and every transverse loop is essential in $\pi_1\left(M\right)$. {{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.37<br />
<br />
A taut foliation has no Reeb component. If $M$ is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.<br />
<br />
{{beginthm|Theorem|(Palmeira)}} If $\mathcal{F}$ is a taut foliation of a 3-manifold $M$ not finitely covered by $S^2\times S^1$, then the universal covering $\widetilde{M}$ is homeomorphic to ${\mathbb R}^3$ and the pull-back foliation $\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$ is homeomorphic to a product foliation $\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$, where $\mathcal{G}$ is a foliation of ${\mathbb R}^2$ by lines.{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.38<br />
<br />
In particular, a taut foliation of a 3-manifold $M$ yields an action of $\pi_1M$ on a (possibly non-Hausdorff) simply connected 1-manifold $L$, the space of leaves of ${\mathcal{G}}$.<br />
<br />
{{beginthm|Theorem|(Gabai)}} Let $M$ be a closed, irreducible 3-manifold.<br />
<br />
a) If $H_2\left(M;{\mathbb R}\right)\not =0$, then $M$ admits a taut foliation.<br />
<br />
b) If $S$ is a surface which minimizes the Thurston norm in its homology class $\left[S\right]\in H_2\left(M;{\mathbb R}\right)$, then $M$ admits a taut foliaton for which $S$ is a leaf.{{endthm}}<br />
<br />
=== Codimension two foliations ===<br />
<br />
==== $S^1$-foliations of 3-manifolds ====<br />
<br />
{{beginthm|Theorem|(Epstein)}} Every foliation of a compact 3-manifold by circles is a Seifert fibration.{{endthm}}<br />
<br />
{{beginthm|Example}}<br />
a) For every rational number $\frac{p}{q}\not=0$ there exists a foliaton of $S^3$ by circles such that restriction to the standard embedded torus is a rational foliation of slope $\frac{p}{q}$. <br />
b) The complement of a knot $K\subset S^3$ admits a foliation by circles if and only if $K$ is a torus knot.{{endthm}}<br />
<br />
{{beginthm|Theorem|(Vogt)}} If a 3-manifold $M$ admits a foliation by circles, then any 3-manifold obtained by removing finitely many points from $M$ admits a (not necessarily smooth) foliation by circles.{{endthm}}<br />
<br />
{{beginthm|Corollary}} ${\mathbb R}^3$ admits a foliation by circles.{{endthm}}<br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
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<br />
[[Category:Manifolds]]<br />
{{Stub}}</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationsFoliations2010-06-14T09:32:48Z<p>Kuessner: </p>
<hr />
<div><!-- COMMENT: <br />
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To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
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- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
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{{Stub}}<br />
<br />
== Introduction ==<br />
<wikitex>;<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
=== Bundles ===<br />
<br />
<wikitex>;<br />
<br />
The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)<br />
<br />
A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth) manifold $M$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)<br />
<br />
Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a diffeomorphism $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:<br />
$$\begin{xy}<br />
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\<br />
U\ar[r]^{id}&U}<br />
\end{xy}$$<br />
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.<br />
</wikitex><br />
<br />
=== Suspensions ===<br />
<wikitex>;<br />
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=<br />
\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\mbox{\ for\ }f\in F,$$<br />
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.<br />
<br />
In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.<br />
<br />
The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. <br />
If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. <br />
</wikitex><br />
<br />
=== Submersions ===<br />
<wikitex>;<br />
Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. <br />
[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]<br />
An example of a submersion, which is not a fiber bundle, is given by<br />
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$<br />
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$<br />
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.<br />
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$<br />
for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the M&ouml;bius band). Their leaf spaces are not Hausdorff.<br />
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]<br />
</wikitex><br />
<br />
=== Reeb foliations ===<br />
<wikitex>;<br />
Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by<br />
$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$<br />
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ <br />
for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.<br />
<br />
</wikitex><br />
<br />
=== Taut foliations ===<br />
<wikitex>;<br />
A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. <br />
<br />
{{beginthm|Theorem|(Rummler, Sullivan)}}<br />
The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$:<br />
<br />
a) $\mathcal{F}$ is taut;<br />
<br />
b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$;<br />
<br />
c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}} <br />
<br />
<br />
</wikitex><br />
<br />
=== Constructing new foliations from old ones ===<br />
<br />
==== Pullbacks ====<br />
<wikitex>;<br />
{{beginthm|Theorem |}} If $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ and $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$, then $N$ is foliated by connected components of $f^{-1}\left(L\right)$ as $L$ ranges over the leaves of $\mathcal{F}$. {{endthm}}<br />
{{cite|Candel&Conlon2000}}, Theorem 3.2.2<br />
</wikitex><br />
<br />
==== Glueing ====<br />
<wikitex>;<br />
Let $\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$ be $n$-manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$.<br />
If either both foliations are tangent or both foliations are transverse to the boundaries of $M_1$ and $M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$. This is called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$.<br />
</wikitex><br />
<br />
==== Turbulization ====<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$. <br />
<br />
Define a foliation ${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by<br />
$$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$<br />
where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$<br />
The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhood of the boundary of $N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$ of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<br />
=== Holonomy ===<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation and $L$ a leaf. For a path $\gamma:\left[0,1\right]\rightarrow L$ contained in the intersection of the leaf $L$ with<br />
a foliation chart $U$, and two transversals $\tau_0,\tau_1$ to $\gamma$ at the endpoints, the product structure of the foliation chart determines a homeomorphism $$h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$$<br />
If $\gamma$ is covered by foliation charts $U_0,\ldots,U_k$, then one obtains a sequence of homeomorphisms $$h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$$<br />
The composition yields a well-defined map $h$ from the germ of $\tau_0$ at $\gamma\left(0\right)$<br />
to the germ of $\tau_{k+1}$ at $\gamma\left(1\right)$, the so-called holonomy transport. The holonomy transport only depends on the relative homotopy class of $\gamma$. <br />
{{beginthm|Lemma}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation, $L$ a leaf, $x\in L$ and $\tau$ a transversal at $x$. Holonomy transport defines a homomorphism $$H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)$$ to the group of germs of homeomorphisms of $\tau$.<br />
{{endthm}}<br />
<br />
{{beginthm|Corollary|(Reeb)}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension one foliation of a 3-manifold such that some leaf $L$ is homeomorphic to $S^2$. Then $M=S^2\times S^1$ and $\mathcal{F}$ is the product foliation by spheres.<br />
{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.5<br />
<br />
<br />
</wikitex><br />
=== Godbillon-Vey invariant ===<br />
<wikitex>;<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation of a manifold $M$, then its tangential plane field $E$ is defined by a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ and $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. The Godbillon-Vey invariant of $\mathcal{F}$ is defined as<br />
$$gv\left({\mathcal{F}}\right):=\left[\eta\wedge\left(d\eta\right)^q\right]\in H^{2q+1}_{dR}\left(M\right).$$<br />
<br />
The Godbillion-Vey invariant is related to resilience of leaves. A leaf is said to be resilient if it is not properly embedded and its holonomy is not trivial.<br />
<br />
{{beginthm|Theorem|(Duminy)}}<br />
If $\left(M,{\mathcal{F}}\right)$ is a foliation of codimension one and no leaf is resilient, then $gv\left({\mathcal{F}}\right)=0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== Classification ==<br />
<br />
=== Codimension one foliations ===<br />
<br />
==== Existence ====<br />
<wikitex>;<br />
{{beginthm|Theorem}}<br />
A closed smooth manifold $M^n$ has a smooth codimension one foliation if and only if $\chi(M^n)=0$, where $\chi$ denotes the Euler characteristic. <br />
<br />
If $\chi(M^n)=0$, then every $(n-1)$-plane field $\tau^{n-1}$ on $M^n$ is homotopic to the tangent plane field of a smooth codimension one foliation.{{endthm}}<br />
<br />
{{cite|Thurston1976}}<br />
</wikitex><br />
<br />
==== Foliations of surfaces ====<br />
<wikitex>;<br />
If $\left({\mathbb R}^2,{\mathcal{F}}\right)$ is a codimension one foliation of the plane ${\mathbb R}^2$, then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold $L$. This provides a 1-1-correspondence between foliations of ${\mathbb R}^2$ and simply connected 1-manifolds. <br />
<br />
Codimension one foliations on compact surfaces $S$ exist only if $\chi\left(S\right)=0$, that is on the Torus, the Klein bottle, the annulus and the Möbius band.<br />
<br />
A foliation $\left(S,{\mathcal{F}}\right)$ is said to contain a Reeb component resp. a non-orientable Reeb component if the restriction of ${\mathcal{F}}$ to some subsurface $S^\prime\subset S$ is a Reeb foliation resp. a non-orientable Reeb foliation. (This implies that $S^\prime$ is an annulus resp. a Möbius band.)<br />
{{beginthm|Theorem}} <br />
<br />
a) Let $\left(S,{\mathcal{F}}\right)$ be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:<br />
<br />
(1) $\mathcal{F}$ is the suspension of a homeomorphism $f:S^1\rightarrow S^1$ or<br />
<br />
(2) $\mathcal{F}$ contains a Reeb component (orientable or not).<br />
<br />
b) Every foliation of the annulus $S^1\times I$ tangent to the boundary is obtained by glueing together a finite number of Reeb components and a finite number of suspensions<br />
<br />
c) Every foliation of the Möbius band tangent to the boundary is one of the following three possibly glued together with a foliation on $S^1\times I$:<br />
<br />
(1) the non-orientable Reeb component<br />
<br />
(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution<br />
<br />
(3) the suspension of an orientation-reversing homeomorpism $f:I\rightarrow I$.{{endthm}}<br />
{{cite|Hector&Hirsch1981}}, Theorem 4.2.15 and Proposition 4.3.2<br />
</wikitex><br />
<br />
==== Foliations of 3-manifolds ====<br />
<wikitex>;<br />
{{beginthm|Theorem|(Novikov)}} If a 3-manifold $M$ admits a foliation $\mathcal{F}$ without Reeb components, then $\pi_2\left(M\right)=0$, every leaf of $\mathcal{F}$ is incompressible, and every transverse loop is essential in $\pi_1\left(M\right)$. {{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.37<br />
<br />
A taut foliation has no Reeb component. If $M$ is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.<br />
<br />
{{beginthm|Theorem|(Palmeira)}} If $\mathcal{F}$ is a taut foliation of a 3-manifold $M$ not finitely covered by $S^2\times S^1$, then the universal covering $\widetilde{M}$ is homeomorphic to ${\mathbb R}^3$ and the pull-back foliation $\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$ is homeomorphic to a product foliation $\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$, where $\mathcal{G}$ is a foliation of ${\mathbb R}^2$ by lines.{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.38<br />
<br />
In particular, a taut foliation of a 3-manifold $M$ yields an action of $\pi_1M$ on a (possibly non-Hausdorff) simply connected 1-manifold $L$, the space of leaves of ${\mathcal{G}}$.<br />
<br />
{{beginthm|Theorem|(Gabai)}} Let $M$ be a closed, irreducible 3-manifold.<br />
<br />
a) If $H_2\left(M;{\mathbb R}\right)\not =0$, then $M$ admits a taut foliation.<br />
<br />
b) If $S$ is a surface which minimizes the Thurston norm in its homology class $\left[S\right]\in H_2\left(M;{\mathbb R}\right)$, then $M$ admits a taut foliaton for which $S$ is a leaf.{{endthm}}<br />
<br />
=== Codimension two foliations ===<br />
<br />
==== $S^1$-foliations of 3-manifolds ====<br />
<br />
{{beginthm|Theorem|(Epstein)}} Every foliation of a compact 3-manifold by circles is a Seifert fibration.{{endthm}}<br />
<br />
{{beginthm|Theorem|(Vogt)}} If a 3-manifold $M$ admits a foliation by circles, then any 3-manifold obtained by removing finitely many points from $M$ admits a (not necessarily smooth) foliation by circles.{{endthm}}<br />
<br />
{{beginthm|Corollary}} ${\mathbb R}^3$ admits a foliation by circles.{{endthm}}<br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
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<br />
[[Category:Manifolds]]<br />
{{Stub}}</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Self-maps_of_simply_connected_manifoldsSelf-maps of simply connected manifolds2010-06-10T11:19:43Z<p>Kuessner: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
== Question ==<br />
<wikitex>;<br />
Let us call an oriented closed connected manifold '''flexible''' if it admits a self-map <br />
that has non-trivial degree (i.e., degree not equal to 1, 0, or -1).<br />
<br />
{{beginthm|Question}}<br />
Do there exist closed simply connected manifolds (of non-zero dimension) <br />
that are not flexible?<br />
{{endthm}}<br />
<br />
{{beginthm|Remark}}<br />
In the following, for simplicity, we implicitly assume that all manifolds are of non-zero dimension.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== Examples and partial answers == <br />
=== Examples of flexible manifolds ===<br />
<wikitex>;<br />
* Of course, all spheres are flexible. <br />
* All odd-dimensional real projective spaces are flexible; all complex projective spaces are flexible. <br />
* Products of flexible manifolds with oriented closed connected manifolds are flexible; in particular, tori are flexible.<br />
* All closed simply connected 3-manifolds are flexible.<br />
* All closed simply connected 4-manifolds are flexible {{cite|Duan&Wang2004|Corollary&nbsp;2}}.<br />
* ...<br />
</wikitex><br />
<br />
=== Examples of manifolds that are not flexible ===<br />
<wikitex>;<br />
Notice that there are many oriented closed connected manifolds that are <br />
not flexible. <br />
<br />
{{beginthm|Example}}<br />
* All oriented closed connected manifolds with non-zero [[simplicial volume]] are not flexible (because the simplicial volume is [[Simplicial volume#Functoriality and elementary examples|functorial]]). <br />
* This includes, for instance, oriented closed connected manifolds of non-positive sectional curvature. (More examples and explanations of these facts can be found on the page on [[simplicial volume]].)<br />
{{endthm}}<br />
<br />
However, by a theorem of Gromov, the simplicial volume of closed simply connected manifolds is always zero. So the simplicial volume cannot be used to discover closed simply connected manifolds that are not flexible.<br />
</wikitex><br />
<br />
== Solution ==<br />
<wikitex>;<br />
This problem was in fact solved by Arkowitz and Lupton {{cite|Arkowitz&Lupton2000|Examples 5.1 & 5.2}}: ''There do exist inflexible closed simply connected manifolds''. <br />
<br />
Arkowitz and Lupton give examples of simply connected rational Poincaré differential graded algebras, $\mathcal{M}_1$ and $\mathcal{M}_2$ of dimensions 208 and 228. Both of these algebras have a finite set of homotopy classes of self-maps and so the corresponding rational Poincaré complexes are inflexible. They also indicate why the algebras $\mathcal{M}_1$ and $\mathcal{M}_2$ can be realised by closed simply connected manifolds $M_1$ and $M_2$. It follows that the manifolds $M_1$ and $M_2$ are inflexible.<br />
<br />
The realisation of $\mathcal{M}_1$ and $\mathcal{M}_2$ relies upon a theorem proven independently by Barge and Sullivan. A special case of this theorem is as follows:<br />
<br />
{{beginthm|Theorem|{{cite|Barge1976|Theorem 1}}{{cite|Sullivan1977|Theorem 13.2}}}}<br />
Let $\mathcal{M}$ be a simply connected rational Poincaré differential graded algebra of dimension n. If either<br />
* $n \neq 4k$ or<br />
* $n = 4k$ and the intersection form of $\mathcal{M}$ represents the trivial element of $W_0(\Qq)$, the Witt group of $\Qq$, <br />
then there is a closed simply connected smooth manifold $M$ with rational homotopy type given by $\mathcal{M}$. <br />
{{endthm}}<br />
<br />
{{beginthm|Remark}}<br />
Note that Arkowitz and Lupton {{cite|Arkowitz&Lupton2000}} state that the theorem of Barge-Sullivan applies and do not give detailed arguments why $\mathcal{M}_1$ or $\mathcal{M}_2$ satisfy the hypotheses of Barge-Sullivan. See the [[Talk:Self-maps of simply connected manifolds#Details for Arkowitz and Lupton's paper|discussion page]] for more information.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please add further headings according to your needs. --><br />
<br />
[[Category:Questions]]<br />
[[Category:Research questions]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Talk:Lie_groups_I:_Definition_and_examplesTalk:Lie groups I: Definition and examples2010-06-09T10:59:35Z<p>Kuessner: </p>
<hr />
<div>Theorem 2.2 is wrong as stated. A counterexample is given on page 83 of {{cite|Carter&Segal&Macdonald1995}}.<br />
<br />
For each Lie group one has the adjoint representation $Ad:G\rightarrow GL\left(g\right)$. This representation is faithful if $G$ is semisimple. Thus Theorem 2.2 is correct for semisimple Lie groups.<br />
<br />
I'm going to add the condition "G semisimple" to the assumptions of Theorem 2.2.<br />
<br />
<br />
Nachtrag: Now that the statement has changed to include compact Lie groups only: wouldn't it be clearer to state that compact Lie groups are isomorphic to subgroups of O(n) not just of GL(n,R)?<br />
<br />
Or perhaps one should divide the theorem into two parts:<br />
a) each semisimple Lie group is isomorphic to a subgroup of GL(n,R)<br />
b) each compact. Lie group is isomorphic to a subgroup of O(n)</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/FoliationsFoliations2010-06-09T09:08:03Z<p>Kuessner: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
--><br />
{{Stub}}<br />
<br />
== Introduction ==<br />
<wikitex>;<br />
Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.<br />
<br />
The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$.<br />
<br />
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$<br />
A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.<br />
<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$,<br />
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$<br />
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$.<br />
<br />
The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
=== Bundles ===<br />
<br />
<wikitex>;<br />
<br />
The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)<br />
<br />
A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth) manifold $M$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)<br />
<br />
Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a diffeomorphism $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:<br />
$$\begin{xy}<br />
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\<br />
U\ar[r]^{id}&U}<br />
\end{xy}$$<br />
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.<br />
</wikitex><br />
<br />
=== Suspensions ===<br />
<wikitex>;<br />
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=<br />
\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\mbox{\ for\ }f\in F,$$<br />
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.<br />
<br />
In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.<br />
<br />
The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. <br />
If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. <br />
</wikitex><br />
<br />
=== Submersions ===<br />
<wikitex>;<br />
Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. <br />
[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]<br />
An example of a submersion, which is not a fiber bundle, is given by<br />
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$<br />
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$<br />
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.<br />
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$<br />
for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the M&ouml;bius band). Their leaf spaces are not Hausdorff.<br />
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]<br />
</wikitex><br />
<br />
=== Reeb foliations ===<br />
<wikitex>;<br />
Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by<br />
$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$<br />
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ <br />
for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.<br />
<br />
</wikitex><br />
<br />
=== Taut foliations ===<br />
<wikitex>;<br />
A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. <br />
<br />
{{beginthm|Theorem|(Rummler, Sullivan)}}<br />
The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$:<br />
<br />
a) $\mathcal{F}$ is taut;<br />
<br />
b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$;<br />
<br />
c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}} <br />
<br />
<br />
</wikitex><br />
<br />
=== Constructing new foliations from old ones ===<br />
<br />
==== Pullbacks ====<br />
<wikitex>;<br />
{{beginthm|Theorem |}} If $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ and $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$, then $N$ is foliated by connected components of $f^{-1}\left(L\right)$ as $L$ ranges over the leaves of $\mathcal{F}$. {{endthm}}<br />
{{cite|Candel&Conlon2000}}, Theorem 3.2.2<br />
</wikitex><br />
<br />
==== Glueing ====<br />
<wikitex>;<br />
Let $\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$ be $n$-manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$.<br />
If either both foliations are tangent or both foliations are transverse to the boundaries of $M_1$ and $M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$. This is called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$.<br />
</wikitex><br />
<br />
==== Turbulization ====<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$. <br />
<br />
Define a foliation ${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by<br />
$$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$<br />
where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$<br />
The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhood of the boundary of $N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$ of $\mathcal{F}$.<br />
</wikitex><br />
<br />
== Invariants ==<br />
<br />
=== Holonomy ===<br />
<wikitex>;<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation and $L$ a leaf. For a path $\gamma:\left[0,1\right]\rightarrow L$ contained in the intersection of the leaf $L$ with<br />
a foliation chart $U$, and two transversals $\tau_0,\tau_1$ to $\gamma$ at the endpoints, the product structure of the foliation chart determines a homeomorphism $$h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$$<br />
If $\gamma$ is covered by foliation charts $U_0,\ldots,U_k$, then one obtains a sequence of homeomorphisms $$h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$$<br />
The composition yields a well-defined map $h$ from the germ of $\tau_0$ at $\gamma\left(0\right)$<br />
to the germ of $\tau_{k+1}$ at $\gamma\left(1\right)$, the so-called holonomy transport. The holonomy transport only depends on the relative homotopy class of $\gamma$. <br />
{{beginthm|Lemma}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a foliation, $L$ a leaf, $x\in L$ and $\tau$ a transversal at $x$. Holonomy transport defines a homomorphism $$H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)$$ to the group of germs of homeomorphisms of $\tau$.<br />
{{endthm}}<br />
<br />
{{beginthm|Corollary|(Reeb)}}<br />
Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension one foliation of a 3-manifold such that some leaf $L$ is homeomorphic to $S^2$. Then $M=S^2\times S^1$ and $\mathcal{F}$ is the product foliation by spheres.<br />
{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.5<br />
<br />
<br />
</wikitex><br />
=== Godbillon-Vey invariant ===<br />
<wikitex>;<br />
If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation of a manifold $M$, then its tangential plane field $E$ is defined by a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ and $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. The Godbillon-Vey invariant of $\mathcal{F}$ is defined as<br />
$$gv\left({\mathcal{F}}\right):=\left[\eta\wedge\left(d\eta\right)^q\right]\in H^{2q+1}_{dR}\left(M\right).$$<br />
<br />
The Godbillion-Vey invariant is related to resilience of leaves. A leaf is said to be resilient if it is not properly embedded and its holonomy is not trivial.<br />
<br />
{{beginthm|Theorem|(Duminy)}}<br />
If $\left(M,{\mathcal{F}}\right)$ is a foliation of codimension one and no leaf is resilient, then $gv\left({\mathcal{F}}\right)=0$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
== Classification ==<br />
<br />
=== Codimension one foliations ===<br />
<br />
==== Existence ====<br />
<wikitex>;<br />
{{beginthm|Theorem}}<br />
A closed smooth manifold $M^n$ has a smooth codimension one foliation if and only if $\chi(M^n)=0$, where $\chi$ denotes the Euler characteristic. <br />
<br />
If $\chi(M^n)=0$, then every $(n-1)$-plane field $\tau^{n-1}$ on $M^n$ is homotopic to the tangent plane field of a smooth codimension one foliation.{{endthm}}<br />
<br />
{{cite|Thurston1976}}<br />
</wikitex><br />
<br />
==== Foliations of surfaces ====<br />
<wikitex>;<br />
If $\left({\mathbb R}^2,{\mathcal{F}}\right)$ is a codimension one foliation of the plane ${\mathbb R}^2$, then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold $L$. This provides a 1-1-correspondence between foliations of ${\mathbb R}^2$ and simply connected 1-manifolds. <br />
<br />
Codimension one foliations on compact surfaces $S$ exist only if $\chi\left(S\right)=0$, that is on the Torus, the Klein bottle, the annulus and the Möbius band.<br />
<br />
A foliation $\left(S,{\mathcal{F}}\right)$ is said to contain a Reeb component resp. a non-orientable Reeb component if the restriction of ${\mathcal{F}}$ to some subsurface $S^\prime\subset S$ is a Reeb foliation resp. a non-orientable Reeb foliation. (This implies that $S^\prime$ is an annulus resp. a Möbius band.)<br />
{{beginthm|Theorem}} <br />
<br />
a) Let $\left(S,{\mathcal{F}}\right)$ be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:<br />
<br />
(1) $\mathcal{F}$ is the suspension of a homeomorphism $f:S^1\rightarrow S^1$ or<br />
<br />
(2) $\mathcal{F}$ contains a Reeb component (orientable or not).<br />
<br />
b) Every foliation of the annulus $S^1\times I$ tangent to the boundary is obtained by glueing together a finite number of Reeb components and a finite number of suspensions<br />
<br />
c) Every foliation of the Möbius band tangent to the boundary is one of the following three possibly glued together with a foliation on $S^1\times I$:<br />
<br />
(1) the non-orientable Reeb component<br />
<br />
(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution<br />
<br />
(3) the suspension of an orientation-reversing homeomorpism $f:I\rightarrow I$.{{endthm}}<br />
{{cite|Hector&Hirsch1981}}, Theorem 4.2.15 and Proposition 4.3.2<br />
</wikitex><br />
<br />
==== Foliations of 3-manifolds ====<br />
<wikitex>;<br />
{{beginthm|Theorem|(Novikov)}} If a 3-manifold $M$ admits a foliation $\mathcal{F}$ without Reeb components, then $\pi_2\left(M\right)=0$, every leaf of $\mathcal{F}$ is incompressible, and every transverse loop is essential in $\pi_1\left(M\right)$. {{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.37<br />
<br />
A taut foliation has no Reeb component. If $M$ is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.<br />
<br />
{{beginthm|Theorem|(Palmeira)}} If $\mathcal{F}$ is a taut foliation of a 3-manifold $M$ not finitely covered by $S^2\times S^1$, then the universal covering $\widetilde{M}$ is homeomorphic to ${\mathbb R}^3$ and the pull-back foliation $\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$ is homeomorphic to a product foliation $\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$, where $\mathcal{G}$ is a foliation of ${\mathbb R}^2$ by lines.{{endthm}}<br />
{{cite|Calegari2007}} Theorem 4.38<br />
<br />
In particular, a taut foliation of a 3-manifold $M$ yields an action of $\pi_1M$ on a (possibly non-Hausdorff) simply connected 1-manifold $L$, the space of leaves of ${\mathcal{G}}$.<br />
<br />
{{beginthm|Theorem|(Gabai)}} Let $M$ be a closed, irreducible 3-manifold.<br />
<br />
a) If $H_2\left(M;{\mathbb R}\right)\not =0$, then $M$ admits a taut foliation.<br />
<br />
b) If $S$ is a surface which minimizes the Thurston norm in its homology class $\left[S\right]\in H_2\left(M;{\mathbb R}\right)$, then $M$ admits a taut foliaton for which $S$ is a leaf.{{endthm}}<br />
<br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
{{Stub}}</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Geometric_3-manifoldsGeometric 3-manifolds2010-06-09T09:07:34Z<p>Kuessner: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
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END OF COMMENT<br />
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--><br />
{{Stub}}<br />
<br />
== Introduction ==<br />
<wikitex>;<br />
Let a group $G$ act on a manifold $X$ by homeomorphisms. <br />
<br />
A $\left(G,X\right)$-manifold is a manifold $M$ with a $\left(G,X\right)$-atlas, that is, a collection $\left\{\left(U_i,\phi_i\right):i\in I\right\}$ of homeomorphisms $$\phi_i:U_i\rightarrow \phi_i\left(U_i\right)\subset X$$ onto open subsets of $X$ such that all coordinate changes $$\gamma_{ij}=\phi_i\phi_j^{-1}:\phi_i\left(U_i\cap U_j\right)\rightarrow \phi_j\left(U_i\cap U_j\right)$$ are restrictions of elements of $G$.<br />
<br />
Fix a basepoint $x_0\in M$ and a chart $\left(U_0,\phi_0\right)$ with $x_0\in U_0$. Let $\pi:\widetilde{M}\rightarrow M$ be the universal covering. These data determine the developing map $$D:\widetilde{M}\rightarrow X$$ that agrees with the analytic continuation of $\phi_0\pi$ along each path, in a neighborhood of the path's endpoint.<br />
<br />
If we change the initial data $x_0$ and $\left(U_0,\phi_0\right)$, the developing map $D$ changes by composition with an element of $G$.<br />
<br />
If $\sigma\in\pi_1\left(M,x_0\right)$, analytic continuation along a loop representing $\sigma$ gives a chart $\phi_0^\sigma$ that is comparable to $\phi_0$, since they are both defined at $x_0$. Let $g_\sigma$ be the element of $G$ such that $\phi_0^\sigma=g_\sigma\phi_0$. The map $$H:\pi_1\left(M,x_0\right)\rightarrow G, H\left(\sigma\right)=g_\sigma$$<br />
is a group homomorphism and is called the holonomy of $M$.<br />
<br />
If we change the initial data $x_0$ and $\left(U_0,\phi_0\right)$, the holonomy homomorphisms $H$ changes by conjugation with an element of $G$.<br />
<br />
A $\left(G,X\right)$-manifold is complete if the developing map $D:\widetilde{M}\rightarrow X$ is surjective.<br />
<br />
{{cite|Thurston1997}} Section 3.4<br />
<br />
{{beginthm|Definition}} <br />
A model geometry $\left(G,X\right)$ is a smooth manifold $X$ together with a Lie group of diffeomorphisms of $X$, such that:<br />
<br />
a) $X$ is connected and simply connected;<br />
<br />
b) $G$ acts transitively on $X$, with compact point stabilizers;<br />
<br />
c) $G$ is not contained in any larger group of diffeomorphisms of $X$ with compact point stabilizers;<br />
<br />
d) there exists at least one compact $\left(G,X\right)$-manifold.<br />
{{endthm}}<br />
{{cite|Thurston1997}} Definition 3.8.1<br />
<br />
A 3-manifold is said to be a geometric manifold if it is a $\left(G,X\right)$-manifold for a 3-dimensional model geometry $\left(G,X\right)$.<br />
</wikitex><br />
<br />
== Construction and examples ==<br />
<wikitex>;<br />
{{beginthm|Theorem}}There are eight 3-dimensional model geometries:<br />
<br />
- the round sphere: $X=S^3, G=O(4)$<br />
<br />
- Euclidean space: $X={\mathbb R}^3, G={\mathbb R}^3\rtimes O(3)$<br />
<br />
- hyperbolic space: $X= H^3, G=PSL\left(2,{\mathbb C}\right)\rtimes {\mathbb Z}/2{\mathbb Z}$<br />
<br />
- $X=S^2\times {\mathbb R}, G=O(3)\times \left({\mathbb R}\rtimes {\mathbb Z}/2{\mathbb Z}\right)$<br />
<br />
- $X={\mathbb H}^2\times {\mathbb R}, G=\left(PSL\left(2,{\mathbb R}\right)\rtimes{\mathbb Z}/2{\mathbb Z}\right)\times \left({\mathbb R}\rtimes {\mathbb Z}/2{\mathbb Z}\right)$<br />
<br />
- the universal covering of the unit tangent bundle of the hyperbolic plane: $G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$<br />
<br />
- the Heisenberg group: $G=X=Nil=\left\{\left(\begin{matrix}1&x&z\\0&1&y\\0&0&1\end{matrix}\right):x,y,z\in{\mathbb R}\right\}$<br />
<br />
- the 3-dimensional solvable Lie group $G=X=Sol={\mathbb R}^2\rtimes {\mathbb R}$ with conjugation $t\rightarrow\left(\begin{matrix}e^t&0\\0&e^{-t}\end{matrix}\right)$.{{endthm}}<br />
{{cite|Thurston1997}} Section 3.8<br />
<br />
Outline of Proof:<br />
<br />
Let $G^\prime$ be the connected component of the identity of $G$, and let $G_x^\prime$ be the stabiliser of $x\in X$. <br />
$G^\prime$ acts transitively and $G_x^\prime$ is a closed, connected subgroup of $SO\left(3\right)$.<br />
<br />
Case 1: $G_x^\prime=SO\left(3\right)$. Then $X$ has constant sectional curvature. The Cartan Theorem implies that (up to rescaling) $X$ is isometric to one of $S^3, {\mathbb R}^3, H^3$.<br />
<br />
Case 2: $G_x^\prime \simeq SO\left(2\right)$. Let $V$ be the $G^\prime$-invariant vector field such that, for each $x\in X$, the direction of $V_x$ is the rotation axis of $G_x^\prime$. $V$ descends to a vector field on compact $\left(G,X\right)$-manifolds, therefore the flow of $V$ must preserve volume. In our setting this implies that the flow of $V$ acts by isometries. Hence the flowlines define a 1-dimensional foliation ${\mathcal{F}}$ with embedded leaves. The quotient $X/{\mathcal{F}}$ is a 2-dimensional manifold, which inherits a Riemannian metric such that $G^\prime$ acts transitively by isometries. Thus $Y:=X/{\mathcal{F}}$ has constant curvature and is (up to rescaling) isometric to one of $S^2, {\mathbb R}^2, H^2$. $X$ is a pricipal bundle over $Y$ with fiber ${\mathbb R}$ or $S^1$,<br />
The plane field $\tau$ orthogonal to $\mathcal{F}$ has constant curvature, hence it is either a foliation or a contact structure.<br />
<br />
Case 2a: $\tau$ is a foliation. Thus $X$ is a flat bundle over $Y$. $Y$ is one of $S^2, {\mathbb R}^2, H^2$, hence $\pi_1Y=0$, which implies that $X=Y\times {\mathbb R}$.<br />
<br />
Case 2b: $\tau$ is a contact structure. <br />
For $Y=S^2$ one would obtain for $G$ the group of isometries of $S^3$ that preserve the Hopf fibration. This is not a maximal group with compact stabilizers, thus there is no model geometry in this case.<br />
For $Y={\mathbb R}^2$ one obtains $G=X=Nil$. Namely, $G$ is the subgroup of the group of automorphisms of the standard contact structure $dz-xdy=0$ on ${\mathbb R}^3$ consisting of thise automorphisms which are lifts of isometries of the x-y-plane.<br />
For $Y={\mathbb H}^2$ one obtains $G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$.<br />
<br />
Case 3: $G_x^\prime=1$. Then $X=G^\prime/G_x^\prime=G^\prime$ is a Lie group. The only 3-dimensional unimodular Lie group which is not subsumed by one of the previous geometries is $G=Sol$.<br />
<br />
<br />
<br />
<br />
</wikitex><br />
<br />
== Invariants ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<br />
A closed 3-manifold is called:<br />
<br />
- irreducible, if every embedded 2-sphere bounds an embedded 3-ball,<br />
<br />
- geometrically atoroidal, if there is no embedded incompressible torus,<br />
<br />
- homotopically atoroidal, if there is no immersed incompressible torus.<br />
<br />
<wikitex>;<br />
{{beginthm|Theorem|(Geometrization)}}<br />
<br />
Let $M$ be a closed, orientable, irreducible, geometrically atoroidal 3-manifold.<br />
<br />
a) If $M$ is homotopically atoroidal, then it admits an $H^3$-geometry.<br />
<br />
b) If $M$ is not homotopically atoroidal, then it admits (at least) one of the seven non-$H^3$-geometries.<br />
{{endthm}}<br />
<br />
{{beginthm|Example|(Geometrization of mapping tori)}} <br />
<br />
Let $\Phi:\Sigma_g\rightarrow \Sigma_g$ be an orientation-preserving homeomorphism of the surface of genus $g$.<br />
<br />
a) If $g=1$, then the mapping torus $M_\Phi$ satisfies the following:<br />
<br />
1. If $\Phi$ is periodic, then $M_\Phi$ admits an ${\mathbb R}^3$ geometry.<br />
<br />
2. If $\Phi$ is reducible, then $M_\Phi$ contains an embedded incompressible torus.<br />
<br />
3. If $\Phi$ is Anosov, then $M_\Phi$ admits a $Sol$ geometry.<br />
<br />
b) If $g\ge 2$, then the mapping torus $M_\Phi$ satisfies the following:<br />
<br />
1. If $\Phi$ is periodic, then $M_\Phi$ admits an $H^2\times{\mathbb R}$-geometry.<br />
<br />
2. If $\Phi$ is reducible, then $M_\Phi$ contains an embedded incompressible torus.<br />
<br />
3. If $\Phi$ is pseudo-Anosov, then $M_\Phi$ admits an $H^3$-geometry.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]<br />
{{Stub}}</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Lie_groups_I:_Definition_and_examplesLie groups I: Definition and examples2010-06-09T09:05:03Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
This is part I of a series of articles about Lie groups. <br />
We define Lie groups and homomorphisms and give the most important examples. In following pages of this series we define the fundamental invariants like the Dynkin diagram, and report about the classification of compact Lie groups. We also report about exceptional Lie groups. <br />
</wikitex><br />
<br />
== Definition and examples ==<br />
<wikitex>;<br />
{{beginthm|Definition}} A Lie group is a finite dimensional smooth manifold $G$ together with a group structure on $G$, such that the multiplication $G \times G \to G$ and the attaching of an inverse $g \mapsto g^{-1}: G \to G$ are smooth maps.<br />
<br />
A morphism between two Lie groups $G$ and $H$ is a map $f:G \to H$, which at the same time is smooth and a group homomorphism. An isomorphism is a bijective map $f$ such that $f$ and $f^{-1} $ are morphisms. A Lie subgroup is a subgroup $H$ in $G$ such that $H$ is also a smooth submanifold of $G$.{{endthm}}<br />
<br />
It is enough to require that the multiplication is smooth, the smoothness of the inverse map can be derived from this. Obviously the product of two Lie groups or a finite sequence of Lie groups is a Lie group. <br />
<br />
<br />
The simplest examples of Lie groups are countable groups, which with the discrete topology are a $0$-dimensional Lie group. In particular all finite groups are $0$-dimensional Lie groups. The most basic Lie groups of positive dimension are matrix groups. The general linear groups over $R$, $\mathbb C$ or the quternions $\mathbb H$:<br />
$$GL(n;\mathbb R) ; \,\, GL(n;\mathbb C);GL(n;\mathbb H),$$ are Lie groups.<br />
They are smooth manifolds as open subsets of the vector space of all corresponding matrices. Their dimension is $n^2$, $2n^2$ and $4n^2$ resp. (note that Lie groups are real manifolds, this explains the formula for the dimension). $GL(n;\mathbb C)$ is also a complex manifold and one obtains a complex Lie group but we will here only consider real Lie groups.<br />
<br />
All these groups are non-compact (for positive dimensions). They contain compact Lie subgroups given by the orthogonal matrices $O(n)$, the '''orthogonal group''', the hermitian matrices $U(n)$, the '''unitary group''', and the symplectic matrices $Sp(n)$, the '''symplectic group'''. The way to see this is to consider the map $A \mapsto A^t$ from $GL(n;R)$ to the vector space of symmetric matrices and to show that the unit matrix $E$ is a regular value. Thus the preimage of $E$, which is $O(n)$ is a smooth submanifold and so $O(n)$ is a Lie submanifold of $GL(n;\mathbb R)$. Similarly one considers the map $A \mapsto \bar A^t$ from $GL(n;\mathbb C)$ to the skew symmetric matrices over $\mathbb C$ and shows again that $E$ is a regular value. The dimension of $O(n)$ is $$ \dim O(n) = n^2 - \frac{n(n+1)}{2} = \frac{n(n-1)}{2}$$ and <br />
$$\dim U(n) =2n^2-( n + \frac{2n(n-1)}{2}) = n^2.$$<br />
By definition these subgroups are closed subgroups. Since they are bounded subspaces of the vector space of all matrices they are compact. They contain the Lie subgroups of all matrices with determinant $1$, the '''special orthogonal group''' or '''special unitary group''':<br />
$$<br />
SO(n); SU(n).<br />
$$<br />
The first is just the component of $E$ in $O(n)$, whereas the second is the preimage of the regular value $1$ in $S^1$ and so has codimension $1$, implying: <br />
$$\dim SU(n) = n^2-1.$$ The symplectic group $Sp(n)$ is not the group of isometries of a non-degenerate skew-symmetric $\mathbb H$-bilinear form, but the group of the hermitian form on $\mathbb H^n$<br />
$$<x,y> := \bar x_1y_1 + \dots + \bar x_ny_n.<br />
$$<br />
With this definition one proceeds as in the examples above and show that $Sp(n)$ is a Lie group of dimension <br />
$$<br />
\dim Sp(n) = n(2n+1).<br />
$$ The determinant of an element of $Sp(n)$ is $1$, thus we don't consider the groups $SSp(n)$. <br />
<br />
As mentioned above $SO(n)$ is the component of $E$ in $O(n)$. For $n>2$ it's fundamental group is $\mathbb Z/2$ \cite{???} generated by the inclusion $S^1 \cong SO(2) \to SO(n)$. If $G$ is a Lie group then by construction of the universal covering this is a Lie group again. In particular the universal covering of $SO(n)$ for $n>2$ is a Lie group denoted by $Spin(n)$, the '''Spinor group'''. There is a more explicit construction of $Spin(n)$ in terms of Clifford algebras \cite{???}. <br />
<br />
Some of the low dimensional Lie groups above occur in two or more ways, for example $$SO(2) \cong U(1) = S^1, \,\, SU(2) \cong Spin(3) \cong Sp(1) \ = S^3.$$<br />
<br />
A special role in the world of Lie groups play the '''tori''', the Lie groups <br />
$$T^n := (S^1)^n.$$ As we will discuss in a later page, a connected abelian Lie group is isomorphic to a torus.<br />
<br />
Here are a few fundamental examples of non-compact Lie groups:<br />
<br />
1.) The ''Lorentz group'' $O(1,3)$ the group of the isometries of the Minkowski space, the isometries of the form on $\mathbb R^4$ given by $<x,y> := x_1y_1 - x_2y_2 - \dots - x_4y_4$. It's dimension is $6$.<br />
<br />
2.) The Heisenberg Group $H $ consisting of upper $3 \times 3$ matrices with diagonal entries $1$. It's dimension is $3$. <br />
<br />
The following theorems give a rough picture of all Lie groups: <br />
<br />
{{beginthm|Theorem}} Assume that the Lie group $G$ is semisimple. Then $G$ is isomorphic to a Lie subgroup of $GL(n;\mathbb R)$ for some $n$ \cite{???}.<br />
{{endthm}}<br />
Proof: Let $Ad:G\rightarrow GL\left(\underline{g}\right)$ be the [[Wikipedia:Adjoint_representation_of_a_Lie_group|Adjoint representation]] of $G$. If $G$ is semisimple, then the adjoint representation is faithful, thus $G$ is mapped isomorphically to a subgroup of $GL\left(\underline{g}\right)$. <br />
<br />
{{beginthm|Theorem}} A subgroup of a Lie group is a Lie subgroup, if and only if it is closed as a topological subspace \cite{???}. <br />
{{endthm}}<br />
<br />
Thus it's easy to test, whether a subgroup of a Lie group is a Lie subgroup.<br />
<br />
<br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
Why are Lie groups interesting? There are many different answers to this question. Probably everybody will agree that Lie groups, in particular compact Lie groups, give the mathematical language for defining and studying '''symmetries'''. Given a geometric object, say a closed smooth manifold $M$ with a Riemannian metric $g$, then one can consider the group of self isometries $Iso(M,g)$. By a theorem of Myers and Steenrod \cite {???} this is a compact Lie group in such a way that the map $Iso(M,g) \times M \to M$ mapping $(f,x)$ to $f(x)$ is a smooth map \cite{Kobayashi transformation groups ...}. Thus we have a smooth action of $Iso(M,g)$ on $M$. The size of this group is a measure for the symmetry of $(M,g)$. In turn if a compact Lie group $G$ acts smoothly on a closed smooth manifold $M$, then there is a Riemannian metric on $M$ such that $G$ acts by isometries (choose an arbitrary Riemannian metric and average it over $G$ using a Haar measure). If the action is effective (meaning that if $g$ acts trivially, the $g =1$), then $G$ is a subgroup of $Iso(M,g)$. <br />
<br />
Motivated by these considerations one defines an invariant for closed smooth manifolds$M$, the '''degree of symmetry''' which is the largest dimension of a compact Lie group acting effectively on $M$ (or equivalently the largest dimension of $Iso(M,g)$ as $g$ varies over all Riemannian metrics of $M$). The following result distinguishes the spheres and real projective spaces from all other manifolds as the most symmetric ones:<br />
<br />
{{beginthm|Theorem \cite{Frobenius-Birkhoff}}} The degree of symmetry of a clsoed manifold $M$ of dimension $m$ is $\le \frac{m(m+1)}{2}$ (the dimension of $O(m+1)$, which acts on $S^m$), and if the degree of symmetry is $\frac{m(m+1)}{2}$ then $M$ is diffeomorphic to $S^m$ or $\mathbb {RP}^m$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Kuessnerhttp://www.map.mpim-bonn.mpg.de/Lie_groups_I:_Definition_and_examplesLie groups I: Definition and examples2010-06-09T08:48:10Z<p>Kuessner: </p>
<hr />
<div>{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
This is part I of a series of articles about Lie groups. <br />
We define Lie groups and homomorphisms and give the most important examples. In following pages of this series we define the fundamental invariants like the Dynkin diagram, and report about the classification of compact Lie groups. We also report about exceptional Lie groups. <br />
</wikitex><br />
<br />
== Definition and examples ==<br />
<wikitex>;<br />
{{beginthm|Definition}} A Lie group is a finite dimensional smooth manifold $G$ together with a group structure on $G$, such that the multiplication $G \times G \to G$ and the attaching of an inverse $g \mapsto g^{-1}: G \to G$ are smooth maps.<br />
<br />
A morphism between two Lie groups $G$ and $H$ is a map $f:G \to H$, which at the same time is smooth and a group homomorphism. An isomorphism is a bijective map $f$ such that $f$ and $f^{-1} $ are morphisms. A Lie subgroup is a subgroup $H$ in $G$ such that $H$ is also a smooth submanifold of $G$.{{endthm}}<br />
<br />
It is enough to require that the multiplication is smooth, the smoothness of the inverse map can be derived from this. Obviously the product of two Lie groups or a finite sequence of Lie groups is a Lie group. <br />
<br />
<br />
The simplest examples of Lie groups are countable groups, which with the discrete topology are a $0$-dimensional Lie group. In particular all finite groups are $0$-dimensional Lie groups. The most basic Lie groups of positive dimension are matrix groups. The general linear groups over $R$, $\mathbb C$ or the quternions $\mathbb H$:<br />
$$GL(n;\mathbb R) ; \,\, GL(n;\mathbb C);GL(n;\mathbb H),$$ are Lie groups.<br />
They are smooth manifolds as open subsets of the vector space of all corresponding matrices. Their dimension is $n^2$, $2n^2$ and $4n^2$ resp. (note that Lie groups are real manifolds, this explains the formula for the dimension). $GL(n;\mathbb C)$ is also a complex manifold and one obtains a complex Lie group but we will here only consider real Lie groups.<br />
<br />
All these groups are non-compact (for positive dimensions). They contain compact Lie subgroups given by the orthogonal matrices $O(n)$, the '''orthogonal group''', the hermitian matrices $U(n)$, the '''unitary group''', and the symplectic matrices $Sp(n)$, the '''symplectic group'''. The way to see this is to consider the map $A \mapsto A^t$ from $GL(n;R)$ to the vector space of symmetric matrices and to show that the unit matrix $E$ is a regular value. Thus the preimage of $E$, which is $O(n)$ is a smooth submanifold and so $O(n)$ is a Lie submanifold of $GL(n;\mathbb R)$. Similarly one considers the map $A \mapsto \bar A^t$ from $GL(n;\mathbb C)$ to the skew symmetric matrices over $\mathbb C$ and shows again that $E$ is a regular value. The dimension of $O(n)$ is $$ \dim O(n) = n^2 - \frac{n(n+1)}{2} = \frac{n(n-1)}{2}$$ and <br />
$$\dim U(n) =2n^2-( n + \frac{2n(n-1)}{2}) = n^2.$$<br />
By definition these subgroups are closed subgroups. Since they are bounded subspaces of the vector space of all matrices they are compact. They contain the Lie subgroups of all matrices with determinant $1$, the '''special orthogonal group''' or '''special unitary group''':<br />
$$<br />
SO(n); SU(n).<br />
$$<br />
The first is just the component of $E$ in $O(n)$, whereas the second is the preimage of the regular value $1$ in $S^1$ and so has codimension $1$, implying: <br />
$$\dim SU(n) = n^2-1.$$ The symplectic group $Sp(n)$ is not the group of isometries of a non-degenerate skew-symmetric $\mathbb H$-bilinear form, but the group of the hermitian form on $\mathbb H^n$<br />
$$<x,y> := \bar x_1y_1 + \dots + \bar x_ny_n.<br />
$$<br />
With this definition one proceeds as in the examples above and show that $Sp(n)$ is a Lie group of dimension <br />
$$<br />
\dim Sp(n) = n(2n+1).<br />
$$ The determinant of an element of $Sp(n)$ is $1$, thus we don't consider the groups $SSp(n)$. <br />
<br />
As mentioned above $SO(n)$ is the component of $E$ in $O(n)$. For $n>2$ it's fundamental group is $\mathbb Z/2$ \cite{???} generated by the inclusion $S^1 \cong SO(2) \to SO(n)$. If $G$ is a Lie group then by construction of the universal covering this is a Lie group again. In particular the universal covering of $SO(n)$ for $n>2$ is a Lie group denoted by $Spin(n)$, the '''Spinor group'''. There is a more explicit construction of $Spin(n)$ in terms of Clifford algebras \cite{???}. <br />
<br />
Some of the low dimensional Lie groups above occur in two or more ways, for example $$SO(2) \cong U(1) = S^1, \,\, SU(2) \cong Spin(3) \cong Sp(1) \ = S^3.$$<br />
<br />
A special role in the world of Lie groups play the '''tori''', the Lie groups <br />
$$T^n := (S^1)^n.$$ As we will discuss in a later page, a connected abelian Lie group is isomorphic to a torus.<br />
<br />
Here are a few fundamental examples of non-compact Lie groups:<br />
<br />
1.) The ''Lorentz group'' $O(1,3)$ the group of the isometries of the Minkowski space, the isometries of the form on $\mathbb R^4$ given by $<x,y> := x_1y_1 - x_2y_2 - \dots - x_4y_4$. It's dimension is $6$.<br />
<br />
2.) The Heisenberg Group $H $ consisting of upper $3 \times 3$ matrices with diagonal entries $1$. It's dimension is $3$. <br />
<br />
The following theorems give a rough picture of all Lie groups: <br />
<br />
{{beginthm|Theorem}} Assume that the Lie group $G$ is semisimple. Then $G$ is isomorphic to a Lie subgroup of $GL(n;\mathbb R)$ for some $n$ \cite{???}.<br />
{{endthm}}<br />
Proof: Let $Ad:G\rightarrow GL\left(\underline{g}\right)$ be the [[Wikipedia:Adjoint_representation_of_a_Lie_group|Adjoint representation]] of $G$. If $G$ is semisimple, then the adjoint representation is faithful, thus $G$ is mapped isomorphically to a subgroup of $GL\left(\underline{g}\right)$. <br />
<br />
{{beginthm|Theorem}} A subgroup of a Lie group is a Lie subgroup, if and only if it is closed as a topological subspace \cite{???}. <br />
{{endthm}}<br />
<br />
Thus it's easy to test, whether a subgroup of a Lie group is a Lie subgroup.<br />
<br />
<br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
Why are Lie groups interesting? There are many different answers to this question. Probably everybody will agree that Lie groups, in particular compact Lie groups, give the mathematical language for defining and studying '''symmetries'''. Given a geometric object, say a closed smooth manifold $M$ with a Riemannian metric $g$, then one can consider the group of self isometries $Iso(M,g)$. By a theorem of Myers and Steenrod \cite {???} this is a compact Lie group in such a way that the map $Iso(M,g) \times M \to M$ mapping $(f,x)$ to $f(x)$ is a smooth map \cite{Kobayashi transformation groups ...}. Thus we have a smooth action of $Iso(M,g)$ on $M$. The size of this group is a measure for the symmetry of $(M,g)$. In turn if a compact Lie group $G$ acts smoothly on a closed smooth manifold $M$, then there is a Riemannian metric on $M$ such that $G$ acts by isometries (choose an arbitrary Riemannian metric and average it over $G$ using a Haar measure). If the action is effective (meaning that if $g$ acts trivially, the $g =1$), then $G$ is a subgroup of $Iso(M,g)$. <br />
<br />
Motivated by these considerations one defines an invariant for closed smooth manifolds$M$, the '''degree of symmetry''' which is the largest dimension of a compact Lie group acting effectively on $M$ (or equivalently the largest dimension of $Iso(M,g)$ as $g$ varies over all Riemannian metrics of $M$). The following result distinguishes the spheres and real projective spaces from all other manifolds as the most symmetric ones:<br />
<br />
{{beginthm|Theorem \cite{Frobenius-Birkhoff}}} The degree of symmetry of a clsoed manifold $M$ of dimension $m$ is $\le \frac{m(m+1)}{2}$ (the dimension of $O(n+1)$, which acts on $S^n$), and if the degree of symmetry is $\frac{m(m+1)}{2}$ then $M$ is diffeomorphic to $S^m$ or $\mathbb {RP}^m$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
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[[Category:Manifolds]]</div>Kuessner