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1 Introduction

1 Foliations

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).

2 Defining differential form

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).

3 Leaves

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 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}.

4 Holonomy Cocycle

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.

5 Special classes of foliations

5.1 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 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
\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 homeomorphism (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
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, in particular L is a Hausdorff manifold.

1.1 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.

1.2 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

1.3 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.

1.4 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 3.1 (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 Constructing new foliations from old ones

2.1 Pullbacks

Theorem 4.1. 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

2.2 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.

2.3 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 References

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