Foliations

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==== Existence ====
==== Existence ====
<wikitex>;
<wikitex>;
{{beginthm|Theorem}}{{cite|Thurston1976}}
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{{beginthm|Theorem}}
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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}}
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Reference: {{cite|Thurston1976}}
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</wikitex>

Revision as of 16:25, 7 June 2010

Contents

1 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
\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 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 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
\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}} 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,

\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 for some \eta\in\Omega^1\left(M\right).


2 Construction and examples

2.1 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) manifold M. 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).) A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers.

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 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:
\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. Its leaf space is (diffeomeorphic) homeomorphic to B.

2.2 Submersions

Let
\displaystyle 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
\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} which is invariant under the {\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}. 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.

2.3 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} 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
\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}. 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.


2.4 Constructing new foliations from old ones

2.4.1 Pullbacks

Theorem 2.1. 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}.

[Candel&Conlon2000], Theorem 3.2.2

2.4.2 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.

2.4.3 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

\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}, and \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)} 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}.


1 Invariants

1.1 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

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

3 Classification/Characterization

3.1 Codimension one foliations

3.1.1 Existence

Theorem 5.1. 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.

Reference: [Thurston1976]

3.1.2 Foliations of surfaces

Codimension one foliations exist only on surfaces S with \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.)

Theorem 5.2.

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 foliation by fibres of a flat \Homeo\left(S^1\right)-bundle 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 flat \Homeo\left(I\right)-bundles

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) a flat \Homeo\left(I\right)-bundle with monodromy \rho:{\mathbb Z}\rightarrow \Homeo\left(I\right) such that \rho\left(1\right) is orientation-reversing.

[Hector&Hirsch1981], Theorem 4.2.15 and Proposition 4.3.2


4 Further discussion

...

5 References

This page has not been refereed. The information given here might be incomplete or provisional.

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