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{{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> == Construction and examples == === Bundles === <wikitex>; 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 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 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: $$\begin{xy} \xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\ U\ar[r]^{id}&U} \end{xy}$$ 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> === Submersions === <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. [[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 $$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$ $$f\left(x,y\right)=\left(x^2-1\right)e^y.$$ 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. $$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$ 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> === Constructing new foliations from old ones === ==== Pullbacks ==== <wikitex>; {{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}} {{cite|Candel&Conlon2000}}, Theorem 3.2.2 </wikitex> ==== Glueing ==== <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 $f:\partial M_1\rightarrow \partial M_2$. 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> ==== Turbulization ==== <wikitex>; 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 $\mathcal{F}$. 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 $$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$ 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 $$\lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$ 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 == === Holonomy === <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> == Classification == === Codimension one foliations === ==== Existence ==== <wikitex>; {{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. 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}} </wikitex> ==== Foliations of surfaces ==== <wikitex>; 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 Möbius band.) {{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: (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). 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 $S^1\times I$: (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 $f:I\rightarrow I$.{{endthm}} {{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}} === Codimension two foliations === ==== $S^1$-foliations of 3-manifolds ==== {{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}} == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]

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