# Foliations

## 1 Introduction

Let $M$${{Stub}} == Introduction == ; 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}. == Construction and examples == === Bundles === ; 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 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. === Suspensions === ; 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 \pi, then the leaves of {\mathcal{F}}_\alpha are compact. If \alpha is an irrational multiple of \pi, then the leaves of {\mathcal{F}}_\alpha are dense on the 2-torus. === Submersions === ; 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]] === Reeb foliations === ; 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. === Taut foliations === ; 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}} === Constructing new foliations from old ones === ==== Pullbacks ==== ; {{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 ==== Glueing ==== ; 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. ==== Turbulization ==== ; 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}. == Invariants == === Holonomy === ; 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 === Godbillon-Vey invariant === ; 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}} == Classification == === Codimension one foliations === ==== Existence ==== ; {{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}} ==== Foliations of surfaces ==== ; 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 ==== Foliations of 3-manifolds ==== ; {{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 == ; ... == References == {{#RefList:}} [[Category:Manifolds]] {{Stub}}M$ be an $n$$n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$$M$ into connected, topologically immersed submanifolds of dimension $n-q$$n-q$. ${\mathcal{F}}$${\mathcal{F}}$ is said to be a codimension $q$$q$ (smooth) foliation of $M$$M$ if $M$$M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$$\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms
$\displaystyle \phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$
such that for each $\alpha\in{\mathcal{A}}, b\in B$$\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$$F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$$\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$.

The leaves of $\mathcal{F}$$\mathcal{F}$ are the immersed submanifolds $F_b$$F_b$. Each $x\in M$$x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$$\mathcal{F}$ determines its tangential plane field $E\subset TM$$E\subset TM$ by $E_x:=T_xF_b\subset T_xM$$E_x:=T_xF_b\subset T_xM$ if $x\in F_b$$x\in F_b$.

The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$$\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by
$\displaystyle \gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$

A smooth foliation ${\mathcal{F}}$${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$$det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.

If $\mathcal{F}$$\mathcal{F}$ is a smooth, transversely orientable codimension $q$$q$ foliation and $E$$E$ its tangential plane field, then there is a nonsingular $q$$q$-form $\omega\in\Omega^q\left(M\right)$$\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$$x\in M$,

$\displaystyle \omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$

This implies that $d\omega=\omega\wedge\eta$$d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$$\eta\in\Omega^1\left(M\right)$.

The space of leaves is $L=M/\sim$$L=M/\sim$ with the quotient topology, where $x\sim y$$x\sim y$ if and only if $x$$x$ and $y$$y$ belong to the same leaf of $\mathcal{F}$$\mathcal{F}$.

## 2 Construction and examples

### 2.1 Bundles

The most trivial examples of foliations are products $M=B\times F$$M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$$F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$$M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$$B_f:=\left\{f\right\}\times B, f\in F$.)

A more general class are flat $G$$G$-bundles with $G=\Diff\left(F\right)$$G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$$G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$$F$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$$\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$$\Homeo\left(F\right)$-bundle with monodromy $\rho$$\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$$M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$$\pi_1B$ acts on the universal cober $\widetilde{B}$$\widetilde{B}$ by deck transformations and on $F$$F$ by means of the representation $\rho$$\rho$. ($M$$M$ is a flat $\Diff\left(F\right)$$\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$$\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)

Flat bundles fit into the frame work of fiber bundles. A (smooth) map
$\displaystyle \pi:M\rightarrow B$
between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$$b\in B$ has an open neighborhood $U$$U$ such that there is a diffeomorphism $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$$\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$$p_1$ projection to the first factor) commutative:
$\displaystyle \begin{xy} \xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\ U\ar[r]^{id}&U} \end{xy}$
The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$$F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$$L$ is (diffeomeorphic) homeomorphic to
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$B$, in particular $L$$L$ is a Hausdorff manifold.

### 2.2 Suspensions

A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are
$\displaystyle L_f:= \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\mbox{\ for\ }f\in F,$

where $p:\widetilde{B}\times F\rightarrow M$$p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$$\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.

In particular, if $B=S^1$$B=S^1$ and $\phi:F\rightarrow F$$\phi:F\rightarrow F$ is a homeomorphism of $F$$F$, then the suspension foliation of $\phi$$\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$$\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$$\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$$L=F/\sim$, where $x\sim y$$x\sim y$ if $y=\Phi^n\left(x\right)$$y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$$n\in{\mathbb Z}$.

The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$$R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$$\alpha\in\left[0,2\pi\right)$. If $\alpha$$\alpha$ is a rational multiple of $2\pi$$2\pi$, then the leaves of ${\mathcal{F}}_\alpha$${\mathcal{F}}_\alpha$ are compact. If $\alpha$$\alpha$ is an irrational multiple of $2\pi$$2\pi$, then the leaves of ${\mathcal{F}}_\alpha$${\mathcal{F}}_\alpha$ are dense on the 2-torus.

### 2.3 Submersions

Let
$\displaystyle f:M\rightarrow B$
be a submersion. Then $M$$M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$$\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles.
2-dimensional Reeb foliation

An example of a submersion, which is not a fiber bundle, is given by

$\displaystyle f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$
$\displaystyle f\left(x,y\right)=\left(x^2-1\right)e^y.$
This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$$\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$${\mathbb Z}$-actions given by
$\displaystyle z\left(x,y\right)=\left(x,y+z\right)$
resp.
$\displaystyle z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$

for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$$\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$${\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.

3-dimensional Reeb foliation

### 2.4 Reeb foliations

Define a submersion
$\displaystyle f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$
by
$\displaystyle f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$
where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$$\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$$D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$$D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$${\mathbb Z}$-actions given by
$\displaystyle z\left(x,y\right)=\left(x,y+z\right)$

for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$$\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.

### 2.5 Taut foliations

A codimension one foliation $\mathcal{F}$$\mathcal{F}$ of $M$$M$ is taut if for every leaf $\lambda$$\lambda$ of $\mathcal{F}$$\mathcal{F}$ there is a circle transverse to $\mathcal{F}$$\mathcal{F}$ which intersects $\lambda$$\lambda$.

Theorem 2.1 (Rummler, Sullivan). The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$$\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$$M$:

a) $\mathcal{F}$$\mathcal{F}$ is taut;

b) there is a flow transverse to $\mathcal{F}$$\mathcal{F}$ which preserves some volume form on $M$$M$;

c) there is a Riemannian metric on $M$$M$ for which the leaves of $\mathcal{F}$$\mathcal{F}$ are least area surfaces.

### 2.6 Constructing new foliations from old ones

#### 2.6.1 Pullbacks

Theorem 2.2. If $\left(M,{\mathcal{F}}\right)$$\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$$q$ and $f:N\rightarrow M$$f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$$\mathcal{F}$, then $N$$N$ is foliated by connected components of $f^{-1}\left(L\right)$$f^{-1}\left(L\right)$ as $L$$L$ ranges over the leaves of $\mathcal{F}$$\mathcal{F}$.

[Candel&Conlon2000], Theorem 3.2.2

#### 2.6.2 Glueing

Let $\left(M_1,{\mathcal{F}}_1\right)$$\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$$\left(M_2,{\mathcal{F}}_2\right)$ be $n$$n$-manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$$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$$M_1$ and $M_2$$M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$$M_1\cup_f M_2$. This is called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$${\mathcal{F}}_1$ and ${\mathcal{F}}_2$${\mathcal{F}}_2$.

#### 2.6.3 Turbulization

Let $\left(M,{\mathcal{F}}\right)$$\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$$\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$$\mathcal{F}$.

Define a foliation ${\mathcal{F}}_0$${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$$N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by

$\displaystyle cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$
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}$$\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$$\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with
$\displaystyle \lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$

The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$$\mathcal{F}_0$ agree on a neighborhood of the boundary of $N\left(\gamma\left(S^1\right)\right)$$N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$${\mathcal{F}}^\prime$ of $\mathcal{F}$$\mathcal{F}$.

## 3 Invariants

### 3.1 Holonomy

Let $\left(M,{\mathcal{F}}\right)$$\left(M,{\mathcal{F}}\right)$ be a foliation and $L$$L$ a leaf. For a path $\gamma:\left[0,1\right]\rightarrow L$$\gamma:\left[0,1\right]\rightarrow L$ contained in the intersection of the leaf $L$$L$ with

a foliation chart $U$$U$, and two transversals $\tau_0,\tau_1$$\tau_0,\tau_1$ to $\gamma$$\gamma$ at the endpoints, the product structure of the foliation chart determines a homeomorphism
$\displaystyle h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$
If $\gamma$$\gamma$ is covered by foliation charts $U_0,\ldots,U_k$$U_0,\ldots,U_k$, then one obtains a sequence of homeomorphisms
$\displaystyle h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$

The composition yields a well-defined map $h$$h$ from the germ of $\tau_0$$\tau_0$ at $\gamma\left(0\right)$$\gamma\left(0\right)$ to the germ of $\tau_{k+1}$$\tau_{k+1}$ at $\gamma\left(1\right)$$\gamma\left(1\right)$, the so-called holonomy transport. The holonomy transport only depends on the relative homotopy class of $\gamma$$\gamma$.

Lemma 3.1.

Let $\left(M,{\mathcal{F}}\right)$$\left(M,{\mathcal{F}}\right)$ be a foliation, $L$$L$ a leaf, $x\in L$$x\in L$ and $\tau$$\tau$ a transversal at $x$$x$. Holonomy transport defines a homomorphism
$\displaystyle H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)$
to the group of germs of homeomorphisms of $\tau$$\tau$.

Corollary 3.2 (Reeb). Let $\left(M,{\mathcal{F}}\right)$$\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension one foliation of a 3-manifold such that some leaf $L$$L$ is homeomorphic to $S^2$$S^2$. Then $M=S^2\times S^1$$M=S^2\times S^1$ and $\mathcal{F}$$\mathcal{F}$ is the product foliation by spheres.

[Calegari2007] Theorem 4.5

### 3.2 Godbillon-Vey invariant

If $\mathcal{F}$$\mathcal{F}$ is a smooth, transversely orientable codimension $q$$q$ foliation of a manifold $M$$M$, then its tangential plane field $E$$E$ is defined by a nonsingular $q$$q$-form $\omega\in\Omega^q\left(M\right)$$\omega\in\Omega^q\left(M\right)$ and $d\omega=\omega\wedge\eta$$d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$$\eta\in\Omega^1\left(M\right)$. The Godbillon-Vey invariant of $\mathcal{F}$$\mathcal{F}$ is defined as

$\displaystyle gv\left({\mathcal{F}}\right):=\left[\eta\wedge\left(d\eta\right)^q\right]\in H^{2q+1}_{dR}\left(M\right).$

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.

Theorem 3.3 (Duminy). If $\left(M,{\mathcal{F}}\right)$$\left(M,{\mathcal{F}}\right)$ is a foliation of codimension one and no leaf is resilient, then $gv\left({\mathcal{F}}\right)=0$$gv\left({\mathcal{F}}\right)=0$.

## 4 Classification

### 4.1 Codimension one foliations

#### 4.1.1 Existence

Theorem 4.1. A closed smooth manifold $M^n$$M^n$ has a smooth codimension one foliation if and only if $\chi(M^n)=0$$\chi(M^n)=0$, where $\chi$$\chi$ denotes the Euler characteristic.

If $\chi(M^n)=0$$\chi(M^n)=0$, then every $(n-1)$$(n-1)$-plane field $\tau^{n-1}$$\tau^{n-1}$ on $M^n$$M^n$ is homotopic to the tangent plane field of a smooth codimension one foliation.

#### 4.1.2 Foliations of surfaces

If $\left({\mathbb R}^2,{\mathcal{F}}\right)$$\left({\mathbb R}^2,{\mathcal{F}}\right)$ is a codimension one foliation of the plane ${\mathbb R}^2$${\mathbb R}^2$, then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold $L$$L$. This provides a 1-1-correspondence between foliations of ${\mathbb R}^2$${\mathbb R}^2$ and simply connected 1-manifolds.

Codimension one foliations on compact surfaces $S$$S$ exist only if $\chi\left(S\right)=0$$\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)$$\left(S,{\mathcal{F}}\right)$ is said to contain a Reeb component resp. a non-orientable Reeb component if the restriction of ${\mathcal{F}}$${\mathcal{F}}$ to some subsurface $S^\prime\subset S$$S^\prime\subset S$ is a Reeb foliation resp. a non-orientable Reeb foliation. (This implies that $S^\prime$$S^\prime$ is an annulus resp. a Möbius band.)

Theorem 4.2.

a) Let $\left(S,{\mathcal{F}}\right)$$\left(S,{\mathcal{F}}\right)$ be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:

(1) $\mathcal{F}$$\mathcal{F}$ is the suspension of a homeomorphism $f:S^1\rightarrow S^1$$f:S^1\rightarrow S^1$ or

(2) $\mathcal{F}$$\mathcal{F}$ contains a Reeb component (orientable or not).

b) Every foliation of the annulus $S^1\times I$$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$$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$$f:I\rightarrow I$.

[Hector&Hirsch1981], Theorem 4.2.15 and Proposition 4.3.2

#### 4.1.3 Foliations of 3-manifolds

Theorem 4.3 (Novikov). If a 3-manifold $M$$M$ admits a foliation $\mathcal{F}$$\mathcal{F}$ without Reeb components, then $\pi_2\left(M\right)=0$$\pi_2\left(M\right)=0$, every leaf of $\mathcal{F}$$\mathcal{F}$ is incompressible, and every transverse loop is essential in $\pi_1\left(M\right)$$\pi_1\left(M\right)$.

[Calegari2007] Theorem 4.37

A taut foliation has no Reeb component. If $M$$M$ is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.

Theorem 4.4 (Palmeira). If $\mathcal{F}$$\mathcal{F}$ is a taut foliation of a 3-manifold $M$$M$ not finitely covered by $S^2\times S^1$$S^2\times S^1$, then the universal covering
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$\widetilde{M}$ is homeomorphic to ${\mathbb R}^3$${\mathbb R}^3$ and the pull-back foliation $\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$$\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$ is homeomorphic to a product foliation $\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$$\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$, where $\mathcal{G}$$\mathcal{G}$ is a foliation of ${\mathbb R}^2$${\mathbb R}^2$ by lines.

[Calegari2007] Theorem 4.38

In particular, a taut foliation of a 3-manifold $M$$M$ yields an action of $\pi_1M$$\pi_1M$ on a (possibly non-Hausdorff) simply connected 1-manifold $L$$L$, the space of leaves of ${\mathcal{G}}$${\mathcal{G}}$.

Theorem 4.5 (Gabai). Let $M$$M$ be a closed, irreducible 3-manifold.

a) If $H_2\left(M;{\mathbb R}\right)\not =0$$H_2\left(M;{\mathbb R}\right)\not =0$, then $M$$M$ admits a taut foliation.

b) If $S$$S$ is a surface which minimizes the Thurston norm in its homology class $\left[S\right]\in H_2\left(M;{\mathbb R}\right)$$\left[S\right]\in H_2\left(M;{\mathbb R}\right)$, then $M$$M$ admits a taut foliaton for which $S$$S$ is a leaf.

### 1 Codimension two foliations

#### 1.1 $S^1$$S^1$-foliations of 3-manifolds

Theorem 4.6 (Epstein). Every foliation of a compact 3-manifold by circles is a Seifert fibration.

Example 4.7. a) For every rational number $\frac{p}{q}\not=0$$\frac{p}{q}\not=0$ there exists a foliaton of $S^3=\left\{\left(z,w\right)\in{\mathbb C}^2: \mid z\mid^2+\mid w\mid^2=1\right\}$$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\}$$\left\{\left(z,w\right)\in S^3: \mid z\mid=\mid w\mid=1\right\}$ is the suspension foliation of $R_{\frac{p}{q}2\pi}$$R_{\frac{p}{q}2\pi}$.

b) The complement of a knot $K\subset S^3$$K\subset S^3$ admits a foliation by circles if and only if $K$$K$ is a torus knot.
Theorem 4.8 (Vogt). If a 3-manifold $M$$M$ admits a foliation by circles, then any 3-manifold obtained by removing finitely many points from $M$$M$ admits a (not necessarily smooth) foliation by circles.
Corollary 4.9. ${\mathbb R}^3$${\mathbb R}^3$ admits a foliation by circles.

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