Foliations
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=== Submersions === | === Submersions === | ||
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− | Let $$f:M\rightarrow B$$ 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 | + | Let $$f:M\rightarrow B$$ 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. |
+ | [[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]] | ||
+ | An example of a submersion, which is not a fiber bundle, is given by | ||
$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$ | $$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$ | ||
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$ | $$f\left(x,y\right)=\left(x^2-1\right)e^y.$$ |
Revision as of 01:16, 8 June 2010
Contents |
1 Introduction
The leaves of are the submanifolds . Each belongs to a unique leaf. The foliation determines its tangential plane field by if .
The holonomy cocycle of the atlas is given byA smooth foliation is said to be transversely orientable if everywhere.
If is a smooth, transversely orientable codimension foliation and its tangential plane field, then there is a nonsingular -form such that, for each ,
This implies that for some .
2 Construction and examples
2.1 Bundles
The most trivial examples of foliations are products , foliated by the leaves . (Another foliation of is given by .)
A more general class are flat -bundles with or for a (smooth) manifold . Given a representation , the flat -bundle with monodromy is given as , where acts on the universal cober by deck transformations and on by means of the representation . ( is a flat -bundle if .)
Flat bundles fit into the frame work of fiber bundles. A (smooth) mapThe fiber bundle yields a foliation by fibers . Its leaf space is (diffeomeorphic) homeomorphic to .
2.2 Suspensions
where is the canonical projection. This foliation is called the suspension of the representation .
In particular, if and is a homeomorphism of , then the suspension foliation of is defined to be the suspension foliation of the representation given by .
The simplest examples of suspensions are the Kronecker foliations of the 2-torus, that is the suspension foliation of the rotation by angle . If is a rational multiple of , then the leaves of are compact. If is an irrational multiple of , then the leaves of are dense on the 2-torus.
2.3 Submersions
An example of a submersion, which is not a fiber bundle, is given by
for . The induced foliations of 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.
2.4 Reeb foliations
for . The induced foliation of is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.
2.5 Constructing new foliations from old ones
2.5.1 Pullbacks
[Candel&Conlon2000], Theorem 3.2.2
2.5.2 Glueing
Let and be -manifolds with foliations of the same codimension. Assume there is a homeomorphism . If either both foliations are tangent or both foliations are transverse to the boundaries of and , then they can be glued to a foliation on . This is called the tangential resp. the transversal glueing of and .
2.5.3 Turbulization
Let be a transversely orientable codimension 1 foliation, and let be an embedding transverse to .
Define a foliation on a small neighborhood by
The foliations and agree on a neighborhood of the boundary of . The result of glueing these foliations is called the turbulization of .
1 Invariants
1.1 Godbillon-Vey invariant
If is a smooth, transversely orientable codimension foliation of a manifold , then its tangential plane field is defined by a nonsingular -form and for some . The Godbillon-Vey invariant of is defined as
3 Classification/Characterization
3.1 Codimension one foliations
3.1.1 Existence
Theorem 5.1. A closed smooth manifold has a smooth codimension one foliation if and only if , where denotes the Euler characteristic.
If , then every -plane field on is homotopic to the tangent plane field of a smooth codimension one foliation.3.1.2 Foliations of surfaces
Codimension one foliations exist only on surfaces with , that is on the Torus, the Klein bottle, the annulus and the Möbius band.
A foliation is said to contain a Reeb component resp. a non-orientable Reeb component if the restriction of to some subsurface is a Reeb foliation resp. a non-orientable Reeb foliation. (This implies that is an annulus resp. a Möbius band.)
Theorem 5.2.
a) Let be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:
(1) is the suspension of a homeomorphism or
(2) contains a Reeb component (orientable or not).
b) Every foliation of the annulus 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 :
(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 .[Hector&Hirsch1981], Theorem 4.2.15 and Proposition 4.3.2
3.2 Foliations of 3-manifolds
A codimension one foliation of is taut if for every leaf of there is a circle transverse to which intersects . A taut foliation has no Reeb component. If is an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.
Further discussion
...
4 References
- [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
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