Foliations
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This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. | This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. | ||
+ | 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 $\mathcal{F}$. | ||
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U\ar[r]^{id}&U} | U\ar[r]^{id}&U} | ||
\end{xy}$$ | \end{xy}$$ | ||
− | The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its | + | 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 $B$, in particular $L$ is a Hausdorff manifold. |
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where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$. | 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 $\rho\left(z\right)=\Phi^z$. | + | 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 $\alpha\in\left[0,2\pi\right)$. | 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 $\alpha\in\left[0,2\pi\right)$. | ||
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==== Foliations of surfaces ==== | ==== Foliations of surfaces ==== | ||
<wikitex>; | <wikitex>; | ||
− | Codimension one foliations | + | If $\left({\mathbb R}^2,{\mathcal{F}}\right)$ is a codimension one foliation of the plane ${\mathbb R}^2$, then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold $L$. This provides a 1-1-correspondence between foliations of ${\mathbb R}^2$ and simply connected 1-manifolds. |
+ | |||
+ | Codimension one foliations on compact surfaces $S$ exist only if $\chi\left(S\right)=0$, that is on the Torus, the Klein bottle, the annulus and the Möbius 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 Möbius 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 Möbius band.) | ||
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{{beginthm|Theorem|(Palmeira)}} If $\mathcal{F}$ is a taut foliation of a 3-manifold $M$ not finitely covered by $S^2\times S^1$, then the universal covering $\widetilde{M}$ is homeomorphic to ${\mathbb R}^3$ and the pull-back foliation $\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$ is homeomorphic to a product foliation $\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$, where $\mathcal{G}$ is a foliation of ${\mathbb R}^2$ by lines.{{endthm}} | {{beginthm|Theorem|(Palmeira)}} If $\mathcal{F}$ is a taut foliation of a 3-manifold $M$ not finitely covered by $S^2\times S^1$, then the universal covering $\widetilde{M}$ is homeomorphic to ${\mathbb R}^3$ and the pull-back foliation $\left(\widetilde{M},\widetilde{\mathcal{F}}\right)$ is homeomorphic to a product foliation $\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}$, where $\mathcal{G}$ is a foliation of ${\mathbb R}^2$ by lines.{{endthm}} | ||
+ | |||
+ | In particular, a taut foliation of a 3-manifold $M$ yields an action of $\pi_1M$ on a (possibly non-Hausdorff) simply connected 1-manifold $L$, the space of leaves of ${\mathcal{G}}$. | ||
Revision as of 12:10, 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 .
The space of leaves is with the quotient topology, where if and only if and belong to the same leaf of .
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 space of leaves is (diffeomeorphic) homeomorphic to , in particular is a Hausdorff manifold.
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 . Its space of leaves is , where if for some .
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
If is a codimension one foliation of the plane , then its space of leaves is a (possibly non-Hausdorff) simply connected 1-manifold . This provides a 1-1-correspondence between foliations of and simply connected 1-manifolds.
Codimension one foliations on compact surfaces exist only if , 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.
Tex syntax erroris homeomorphic to and the pull-back foliation is homeomorphic to a product foliation , where is a foliation of by lines.
In particular, a taut foliation of a 3-manifold yields an action of on a (possibly non-Hausdorff) simply connected 1-manifold , the space of leaves of .
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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