Foliations
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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 $B_f:=\left\{f\right\}\times B, f\in F$.) | 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 $Diff\left(F\right)$ | + | 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 $\rho\left(\pi_1B\right)\subset Diff\left(F\right)$.) |
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers. | A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers. | ||
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$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$ | $$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$ | ||
$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$ | $$f\left(x,y\right)=\left(x^2-1\right)e^y.$$ | ||
− | This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$- | + | This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp. |
+ | $$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$ | ||
+ | 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 M\"obius band). Their leaf spaces are not Hausdorff. | ||
+ | |||
</wikitex> | </wikitex> | ||
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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. | 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. | ||
+ | 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 M\"obius band.) | ||
{{beginthm|Theorem}} | {{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: | + | 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 | (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). | (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 | 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 | + | |
+ | 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 | (1) the non-orientable Reeb component | ||
+ | |||
(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution | (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}} | (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}} | ||
+ | {{cite|Hector&Hirsch1981}} | ||
</wikitex> | </wikitex> |
Revision as of 16:05, 7 June 2010
Contents |
1 Introduction
Tex syntax error-manifold, possibly with boundary, and let be a decomposition of into connected, topologically immersed submanifolds of dimension . is said to be a codimension foliation of if admits an atlas of foliated charts, that is charts (diffeomorphisms)
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 .) 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
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
2.3.2 Glueing
Tex syntax error-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.
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 5.1. 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.
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
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