Foliations
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=== Codimension one foliations === | === Codimension one foliations === | ||
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+ | === Existence === | ||
<wikitex>; | <wikitex>; | ||
− | A | + | {{beginthm|Theorem}} |
+ | |||
+ | 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}} | ||
+ | {{cite|Thurston1976}} | ||
</wikitex> | </wikitex> | ||
Revision as of 16:18, 7 June 2010
Contents |
1 Introduction
Tex syntax errorbe an -manifold, possibly with boundary, and let be a decomposition of
Tex syntax errorinto connected, topologically immersed submanifolds of dimension . is said to be a codimension foliation of
Tex syntax errorif
Tex syntax erroradmits an atlas of foliated charts, that is charts (diffeomorphisms)
2 Construction and examples
2.1 Bundles
Tex syntax erroris given by .) A more general class are flat -bundles with or for a (smooth) manifold
Tex syntax error. 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 . (
Tex syntax erroris a flat -bundle if .)
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
Tex syntax erroris foliated by the preimages . This includes the case of fiber bundles. 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 M\"obius band). Their leaf spaces are not Hausdorff.
2.3 Constructing new foliations from old ones
2.3.1 Pullbacks
[Candel&Conlon2000], Theorem 3.2.2
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
3.2 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.3.2.1 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.
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 M\"obius 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 foliation by fibres of a flat -bundle 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 flat -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 :
(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 -bundle with monodromy such that is orientation-reversing.[Hector&Hirsch1981], Theorem 4.2.15 and Proposition 4.3.2
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
- [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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