Foliations
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==== Foliations of surfaces ==== | ==== Foliations of surfaces ==== | ||
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− | ... | + | 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 M\"obius band. |
+ | |||
+ | {{beginthm|Theorem}} | ||
+ | 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 fllowing 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.{{endthm}} | ||
+ | |||
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Revision as of 15:52, 7 June 2010
Contents |
1 Introduction
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. 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 . 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) mapThe fiber bundle yields a foliation by fibers . Its leaf space is (diffeomeorphic) homeomorphic to .
2.2 Submersions
This submersion yields a foliation of which is invariant under the -action given by for . The induced foliation of is called the 2-dimensional Reeb foliation. Its leaf space is not Hausdorff.
2.3 Constructing new foliations from old ones
2.3.1 Pullbacks
2.3.2 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 called the tangential resp. the transversal glueing of and .
2.3.3 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 neighborhod of the boundary of . The result of glueing these foliations is called the turbulization of .
Invariants
...
3 Classification/Characterization
3.1 Codimension one foliations
A necessary condition for the existence of a codimension one foliation on a manifold is the vanishing of the Euler characteristic .
3.1.1 Foliations of surfaces
Codimension one foliations exist only on surfaces with , that is on the Torus, the Klein bottle, the annulus and the M\"obius band.
Theorem 5.1.
a) Let \mathcal{F}Homeo\left(S^1\right)\mathcal{F}S^1\times IHomeo\left(I\right)S^1\times IHomeo\left(I\right)\rho:{\mathbb Z}\rightarrow Homeo\left(I\right)\rho\left(1\right)$ is orientation-reversing.
4 Further discussion
...
5 References
- [Candel&Conlon2000] A. Candel and L. Conlon, Foliations. I, American Mathematical Society, Providence, RI, 2000. MR1732868 (2002f:57058) Zbl 0936.57001
This page has not been refereed. The information given here might be incomplete or provisional. |