Foliations

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1 Introduction

Let $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ be an $n$$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

$\displaystyle \phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$ $$\displaystyle \phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$

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

$\displaystyle \gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$ $$\displaystyle \gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$

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$,

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

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

$\displaystyle \pi:M\rightarrow B$ $$\displaystyle \pi:M\rightarrow B$$

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 homeomorphism (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 $B$$B$, in particular $L$$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

$\displaystyle L_f:= \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$ $$\displaystyle L_f:= \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$$

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

$\displaystyle f:M\rightarrow B$ $$\displaystyle f:M\rightarrow B$$

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

$\displaystyle f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$ $$\displaystyle f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$

$\displaystyle f\left(x,y\right)=\left(x^2-1\right)e^y.$ $$\displaystyle f\left(x,y\right)=\left(x^2-1\right)e^y.$$

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

$\displaystyle z\left(x,y\right)=\left(x,y+z\right)$ $$\displaystyle z\left(x,y\right)=\left(x,y+z\right)$$

resp.

$\displaystyle z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$ $$\displaystyle z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$

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

$\displaystyle f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$ $$\displaystyle f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$

by

$\displaystyle f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$ $$\displaystyle f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$

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

$\displaystyle z\left(x,y\right)=\left(x,y+z\right)$ $$\displaystyle z\left(x,y\right)=\left(x,y+z\right)$$

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

$\displaystyle cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$ $$\displaystyle cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$

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

$\displaystyle \lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$ $$\displaystyle \lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$

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

$\displaystyle h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$ $$\displaystyle h:\tau_0\mid_U\rightarrow \tau_1\mid_U.$$

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

$\displaystyle h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$ $$\displaystyle h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.$$

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

$\displaystyle H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)$ $$\displaystyle H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)$$

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

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

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.

[Thurston1976]

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$$S^2\times S^1$, then the universal covering $\widetilde{M}$$\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.

4.2 Codimension two foliations

...

4.2.1 S¹-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.

5 Further discussion

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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