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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 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
\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).

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}.

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).)

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 space of leaves L is (diffeomeorphic) homeomorphic to B, in particular 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 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.

2.3 Submersions

\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.
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} which is invariant under the {\mathbb Z}-actions given by
\displaystyle z\left(x,y\right)=\left(x,y+z\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}. 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.

3-dimensional Reeb foliation

2.4 Reeb foliations

Define a submersion
\displaystyle f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}
\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.

2.5 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.

Theorem 2.1 (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.

2.6 Constructing new foliations from old ones

2.6.1 Pullbacks

Theorem 2.2. 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

2.6.2 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.

2.6.3 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}.

3 Invariants

3.1 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
\displaystyle 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
\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 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.

Lemma 3.1.

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
\displaystyle H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)
to the group of germs of homeomorphisms of \tau.

Corollary 3.2 (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.

[Calegari2007] Theorem 4.5

3.2 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).

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) is a foliation of codimension one and no leaf is resilient, then 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 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.


4.1.2 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.)

Theorem 4.2.

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 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.

[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 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).

[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.

Theorem 4.4 (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
Tex syntax error
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.

[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}}.

Theorem 4.5 (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.

1 Codimension two foliations

1.1 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 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.
Theorem 4.8 (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.
Corollary 4.9. {\mathbb R}^3 admits a foliation by circles.

2 Further discussion


5 References

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

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