Foliations
m |
|||
(42 intermediate revisions by 3 users not shown) | |||
Line 1: | Line 1: | ||
− | < | + | {{Stub}} |
+ | == Introduction == | ||
+ | <wikitex>; | ||
+ | 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 $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$. | ||
− | + | 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 $x\in F_b$. | |
− | + | The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$ | |
− | {{ | + | A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere. |
− | - | + | 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 $x\in M$, |
+ | $$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$ | ||
+ | This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. | ||
− | + | 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 $\mathcal{F}$. | |
− | + | ||
− | + | ||
− | = | + | |
− | + | ||
− | + | ||
</wikitex> | </wikitex> | ||
Line 23: | Line 23: | ||
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 $B_f:=\left\{f\right\}\times B, f\in F$.) | 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 $B_f:=\left\{f\right\}\times B, f\in F$.) | ||
− | A more general class are flat $G$-bundles with $G=Diff\left(F\right)$ or $G=Homeo\left(F\right)$ for a (smooth) manifold $ | + | 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 $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.) |
− | + | ||
− | 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: | + | 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: |
$$\begin{xy} | $$\begin{xy} | ||
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\ | \xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\ | ||
U\ar[r]^{id}&U} | U\ar[r]^{id}&U} | ||
\end{xy}$$ | \end{xy}$$ | ||
− | The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its | + | 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. |
+ | </wikitex> | ||
+ | |||
+ | === Suspensions === | ||
+ | <wikitex>; | ||
+ | A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:= | ||
+ | \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$$ | ||
+ | where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$. | ||
+ | |||
+ | 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 $n\in{\mathbb Z}$. | ||
+ | |||
+ | 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 $\alpha\in\left[0,2\pi\right)$. | ||
+ | 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. | ||
</wikitex> | </wikitex> | ||
=== Submersions === | === Submersions === | ||
<wikitex>; | <wikitex>; | ||
− | 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. An example of a submersion, which is not a fiber bundle, is given by | + | 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. |
+ | [[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]] | ||
+ | An example of a submersion, which is not a fiber bundle, is given by | ||
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$ | $$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$ | ||
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$ | $$f\left(x,y\right)=\left(x^2-1\right)e^y.$$ | ||
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. | 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. | ||
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$ | $$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$ | ||
− | for $\left(x,y\right)\in\left | + | 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. |
+ | [[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]] | ||
+ | </wikitex> | ||
+ | |||
+ | === Reeb foliations === | ||
+ | <wikitex>; | ||
+ | Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by | ||
+ | $$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$ | ||
+ | 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 $$z\left(x,y\right)=\left(x,y+z\right)$$ | ||
+ | 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. | ||
+ | |||
+ | </wikitex> | ||
+ | |||
+ | === Taut foliations === | ||
+ | <wikitex>; | ||
+ | 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 $\lambda$. | ||
+ | |||
+ | {{beginthm|Theorem|(Rummler, Sullivan)}} | ||
+ | The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$: | ||
+ | |||
+ | a) $\mathcal{F}$ is taut; | ||
+ | |||
+ | b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$; | ||
+ | |||
+ | c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}} | ||
+ | |||
</wikitex> | </wikitex> | ||
Line 56: | Line 94: | ||
<wikitex>; | <wikitex>; | ||
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 $f:\partial M_1\rightarrow \partial M_2$. | 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 $f:\partial M_1\rightarrow \partial M_2$. | ||
− | 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 called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$. | + | 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 ${\mathcal{F}}_2$. |
</wikitex> | </wikitex> | ||
− | |||
==== Turbulization ==== | ==== Turbulization ==== | ||
Line 67: | Line 104: | ||
$$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$ | $$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$ | ||
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 $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$ | 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 $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$ | ||
− | The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a | + | 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 $\mathcal{F}$. |
− | + | </wikitex> | |
== Invariants == | == Invariants == | ||
+ | |||
+ | === Holonomy === | ||
<wikitex>; | <wikitex>; | ||
− | ... | + | 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 |
+ | 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 $$h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$$ | ||
+ | If $\gamma$ is covered by foliation charts $U_0,\ldots,U_k$, then one obtains a sequence of homeomorphisms $$h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$$ | ||
+ | The composition yields a well-defined map $h$ from the germ of $\tau_0$ at $\gamma\left(0\right)$ | ||
+ | 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 $\gamma$. | ||
+ | {{beginthm|Lemma}} | ||
+ | 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 $\tau$. | ||
+ | {{endthm}} | ||
+ | |||
+ | {{beginthm|Corollary|(Reeb)}} | ||
+ | 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. | ||
+ | {{endthm}} | ||
+ | {{cite|Calegari2007}} Theorem 4.5 | ||
+ | |||
+ | |||
+ | </wikitex> | ||
+ | === Godbillon-Vey invariant === | ||
+ | <wikitex>; | ||
+ | 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 | ||
+ | $$gv\left({\mathcal{F}}\right):=\left[\eta\wedge\left(d\eta\right)^q\right]\in H^{2q+1}_{dR}\left(M\right).$$ | ||
+ | |||
+ | 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. | ||
+ | |||
+ | {{beginthm|Theorem|(Duminy)}} | ||
+ | If $\left(M,{\mathcal{F}}\right)$ is a foliation of codimension one and no leaf is resilient, then $gv\left({\mathcal{F}}\right)=0$. | ||
+ | {{endthm}} | ||
</wikitex> | </wikitex> | ||
− | == Classification | + | == Classification == |
=== Codimension one foliations === | === Codimension one foliations === | ||
− | === Existence === | + | ==== Existence ==== |
<wikitex>; | <wikitex>; | ||
{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
− | |||
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. | 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. | ||
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}} | 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}} | ||
+ | |||
{{cite|Thurston1976}} | {{cite|Thurston1976}} | ||
</wikitex> | </wikitex> | ||
Line 91: | Line 155: | ||
==== Foliations of surfaces ==== | ==== Foliations of surfaces ==== | ||
<wikitex>; | <wikitex>; | ||
− | Codimension one foliations | + | 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. |
+ | |||
+ | 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. | ||
− | 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 | + | 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.) |
{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
a) Let $\left(S,{\mathcal{F}}\right)$ be a foliated torus or Klein bottle. Then we have one of the two exclusive situations: | a) Let $\left(S,{\mathcal{F}}\right)$ be a foliated torus or Klein bottle. Then we have one of the two exclusive situations: | ||
− | (1) $\mathcal{F}$ is the | + | (1) $\mathcal{F}$ is the suspension of a homeomorphism $f:S^1\rightarrow S^1$ or |
(2) $\mathcal{F}$ contains a Reeb component (orientable or not). | (2) $\mathcal{F}$ contains a Reeb component (orientable or not). | ||
− | 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 | + | 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 |
c) Every foliation of the Möbius band tangent to the boundary is one of the following three possibly glued together with a foliation on $S^1\times I$: | c) Every foliation of the Möbius band tangent to the boundary is one of the following three possibly glued together with a foliation on $S^1\times I$: | ||
Line 110: | Line 176: | ||
(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution | (2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution | ||
− | (3) | + | (3) the suspension of an orientation-reversing homeomorpism $f:I\rightarrow I$.{{endthm}} |
{{cite|Hector&Hirsch1981}}, Theorem 4.2.15 and Proposition 4.3.2 | {{cite|Hector&Hirsch1981}}, Theorem 4.2.15 and Proposition 4.3.2 | ||
+ | </wikitex> | ||
+ | ==== Foliations of 3-manifolds ==== | ||
+ | <wikitex>; | ||
+ | {{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}} | ||
+ | {{cite|Calegari2007}} Theorem 4.37 | ||
+ | |||
+ | A taut foliation has no Reeb component. If $M$ is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut. | ||
+ | |||
+ | {{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}} | ||
+ | {{cite|Calegari2007}} Theorem 4.38 | ||
+ | |||
+ | 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 ${\mathcal{G}}$. | ||
+ | |||
+ | {{beginthm|Theorem|(Gabai)}} Let $M$ be a closed, irreducible 3-manifold. | ||
+ | |||
+ | a) If $H_2\left(M;{\mathbb R}\right)\not =0$, then $M$ admits a taut foliation. | ||
+ | |||
+ | 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}} | ||
+ | </wikitex> | ||
+ | === Codimension two foliations === | ||
+ | <wikitex>; | ||
+ | ... | ||
+ | </wikitex> | ||
+ | ==== S<sup>1</sup>-foliations of 3-manifolds ==== | ||
+ | <wikitex>; | ||
+ | {{beginthm|Theorem|(Epstein)}} Every foliation of a compact 3-manifold by circles is a Seifert fibration.{{endthm}} | ||
+ | |||
+ | {{beginthm|Example}} | ||
+ | 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 $R_{\frac{p}{q}2\pi}$. | ||
+ | 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}} | ||
+ | |||
+ | {{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}} | ||
+ | |||
+ | {{beginthm|Corollary}} ${\mathbb R}^3$ admits a foliation by circles.{{endthm}} | ||
</wikitex> | </wikitex> | ||
Line 123: | Line 223: | ||
{{#RefList:}} | {{#RefList:}} | ||
− | + | == External links == | |
+ | * The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Foliation foliations] | ||
+ | * The Wikipedia page about [[Wikipedia:Foliation|foliations]] | ||
[[Category:Manifolds]] | [[Category:Manifolds]] | ||
− |
Latest revision as of 13:07, 27 March 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
The leaves of are the immersed submanifolds . Each belongs to a unique leaf. The foliation determines its tangential plane field by if .
The holonomy cocycle of the atlas is given byA smooth foliation is said to be transversely orientable if everywhere.
If is a smooth, transversely orientable codimension foliation and its tangential plane field, then there is a nonsingular -form such that, for each ,
This implies that for some .
The space of leaves is with the quotient topology, where if and only if and belong to the same leaf of .
2 Construction and examples
2.1 Bundles
The most trivial examples of foliations are products , foliated by the leaves . (Another foliation of is given by .)
A more general class are flat -bundles with or for a (smooth or topological) manifold . Given a representation , the flat -bundle with monodromy is given as , where acts on the universal cober by deck transformations and on by means of the representation . ( is a flat -bundle if .)
Flat bundles fit into the frame work of fiber bundles. A (smooth) mapThe fiber bundle yields a foliation by fibers . Its space of leaves is (diffeomeorphic) homeomorphic to , in particular is a Hausdorff manifold.
2.2 Suspensions
where is the canonical projection. This foliation is called the suspension of the representation .
In particular, if and is a homeomorphism of , then the suspension foliation of is defined to be the suspension foliation of the representation given by . Its space of leaves is , where if for some .
The simplest examples of suspensions are the Kronecker foliations of the 2-torus, that is the suspension foliation of the rotation by angle . If is a rational multiple of , then the leaves of are compact. If is an irrational multiple of , then the leaves of are dense on the 2-torus.
2.3 Submersions
An example of a submersion, which is not a fiber bundle, is given by
for . The induced foliations of 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.
2.4 Reeb foliations
for . The induced foliation of is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.
2.5 Taut foliations
A codimension one foliation of is taut if for every leaf of there is a circle transverse to which intersects .
Theorem 2.1 (Rummler, Sullivan). The following conditions are equivalent for transversely orientable codimension one foliations of closed, orientable, smooth manifolds :
a) is taut;
b) there is a flow transverse to which preserves some volume form on ;
c) there is a Riemannian metric on for which the leaves of are least area surfaces.
2.6 Constructing new foliations from old ones
2.6.1 Pullbacks
[Candel&Conlon2000], Theorem 3.2.2
2.6.2 Glueing
Let and be -manifolds with foliations of the same codimension. Assume there is a homeomorphism . If either both foliations are tangent or both foliations are transverse to the boundaries of and , then they can be glued to a foliation on . This is called the tangential resp. the transversal glueing of and .
2.6.3 Turbulization
Let be a transversely orientable codimension 1 foliation, and let be an embedding transverse to .
Define a foliation on a small neighborhood by
The foliations and agree on a neighborhood of the boundary of . The result of glueing these foliations is called the turbulization of .
3 Invariants
3.1 Holonomy
Let be a foliation and a leaf. For a path contained in the intersection of the leaf with
a foliation chart , and two transversals to at the endpoints, the product structure of the foliation chart determines a homeomorphismThe composition yields a well-defined map from the germ of at to the germ of at , the so-called holonomy transport. The holonomy transport only depends on the relative homotopy class of .
Lemma 3.1.
Let be a foliation, a leaf, and a transversal at . Holonomy transport defines a homomorphismCorollary 3.2 (Reeb). Let be a transversely orientable codimension one foliation of a 3-manifold such that some leaf is homeomorphic to . Then and is the product foliation by spheres.
[Calegari2007] Theorem 4.5
3.2 Godbillon-Vey invariant
If is a smooth, transversely orientable codimension foliation of a manifold , then its tangential plane field is defined by a nonsingular -form and for some . The Godbillon-Vey invariant of is defined as
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.
Theorem 3.3 (Duminy). If is a foliation of codimension one and no leaf is resilient, then .
4 Classification
4.1 Codimension one foliations
4.1.1 Existence
Theorem 4.1. A closed smooth manifold has a smooth codimension one foliation if and only if , where denotes the Euler characteristic.
If , then every -plane field on is homotopic to the tangent plane field of a smooth codimension one foliation.4.1.2 Foliations of surfaces
If is a codimension one foliation of the plane , then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold . This provides a 1-1-correspondence between foliations of and simply connected 1-manifolds.
Codimension one foliations on compact surfaces exist only if , that is on the Torus, the Klein bottle, the annulus and the Möbius band.
A foliation is said to contain a Reeb component resp. a non-orientable Reeb component if the restriction of to some subsurface is a Reeb foliation resp. a non-orientable Reeb foliation. (This implies that is an annulus resp. a Möbius band.)
Theorem 4.2.
a) Let be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:
(1) is the suspension of a homeomorphism or
(2) contains a Reeb component (orientable or not).
b) Every foliation of the annulus tangent to the boundary is obtained by glueing together a finite number of Reeb components and a finite number of suspensions
c) Every foliation of the Möbius band tangent to the boundary is one of the following three possibly glued together with a foliation on :
(1) the non-orientable Reeb component
(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution
(3) the suspension of an orientation-reversing homeomorpism .[Hector&Hirsch1981], Theorem 4.2.15 and Proposition 4.3.2
4.1.3 Foliations of 3-manifolds
[Calegari2007] Theorem 4.37
A taut foliation has no Reeb component. If is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.
[Calegari2007] Theorem 4.38
In particular, a taut foliation of a 3-manifold yields an action of on a (possibly non-Hausdorff) simply connected 1-manifold , the space of leaves of .
Theorem 4.5 (Gabai). Let be a closed, irreducible 3-manifold.
a) If , then admits a taut foliation.
b) If is a surface which minimizes the Thurston norm in its homology class , then admits a taut foliaton for which is a leaf.4.2 Codimension two foliations
...
4.2.1 S1-foliations of 3-manifolds
Example 4.7. a) For every rational number there exists a foliaton of by circles such that restriction to the standard embedded torus is the suspension foliation of .
b) The complement of a knot admits a foliation by circles if and only if is a torus knot.5 Further discussion
...
6 References
- [Calegari2007] D. Calegari, Foliations and the geometry of 3-manifolds., Oxford Mathematical Monographs; Oxford Science Publications. Oxford University Press, Oxford, 2007. MR2327361 (2008k:57048) Zbl 1118.57002
- [Candel&Conlon2000] A. Candel and L. Conlon, Foliations. I, American Mathematical Society, Providence, RI, 2000. MR1732868 (2002f:57058) Zbl 0936.57001
- [Hector&Hirsch1981] G. Hector and U. Hirsch, Introduction to the geometry of foliations. Part A, Friedr. Vieweg \& Sohn, Braunschweig, 1981. MR639738 (83d:57019) Zbl 0628.57001
- [Thurston1976] W. P. Thurston, Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no.2, 249–268. MR0425985 (54 #13934) Zbl 0347.57014
7 External links
- The Encylopedia of Mathematics article on foliations
- The Wikipedia page about foliations