Foliation

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1 Introduction

1.1 Foliations

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)$.

1.2 Defining differential form

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)$.

1.3 Leaves

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 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}$.

1.4 Holonomy Cocycle

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.

2 Special classes of foliations

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$.