Foliations
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== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
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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 $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth) manifold $ | + | A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$. 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 $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.) |
− | Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a diffeomorphism $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative: | + | Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a homeomorphism (diffeomorphism) $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative: |
$$\begin{xy} | $$\begin{xy} | ||
\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\ | \xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\ | ||
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<wikitex>; | <wikitex>; | ||
A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:= | A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:= | ||
− | \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\mbox{ | + | \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$$ |
where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$. | where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$. | ||
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This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp. | This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp. | ||
$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$ | $$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 | + | 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 Möbius band). Their leaf spaces are not Hausdorff. |
[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]] | [[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]] | ||
</wikitex> | </wikitex> | ||
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b) If $S$ is a surface which minimizes the Thurston norm in its homology class $\left[S\right]\in H_2\left(M;{\mathbb R}\right)$, then $M$ admits a taut foliaton for which $S$ is a leaf.{{endthm}} | b) If $S$ is a surface which minimizes the Thurston norm in its homology class $\left[S\right]\in H_2\left(M;{\mathbb R}\right)$, then $M$ admits a taut foliaton for which $S$ is a leaf.{{endthm}} | ||
+ | </wikitex> | ||
+ | === Codimension two foliations === | ||
+ | <wikitex>; | ||
+ | ... | ||
+ | </wikitex> | ||
+ | ==== S<sup>1</sup>-foliations of 3-manifolds ==== | ||
+ | <wikitex>; | ||
+ | {{beginthm|Theorem|(Epstein)}} Every foliation of a compact 3-manifold by circles is a Seifert fibration.{{endthm}} | ||
+ | {{beginthm|Example}} | ||
+ | a) For every rational number $\frac{p}{q}\not=0$ there exists a foliaton of $S^3=\left\{\left(z,w\right)\in{\mathbb C}^2: \mid z\mid^2+\mid w\mid^2=1\right\}$ by circles such that restriction to the standard embedded torus $\left\{\left(z,w\right)\in S^3: \mid z\mid=\mid w\mid=1\right\}$ is the suspension foliation of $R_{\frac{p}{q}2\pi}$. | ||
+ | b) The complement of a knot $K\subset S^3$ admits a foliation by circles if and only if $K$ is a torus knot.{{endthm}} | ||
+ | |||
+ | {{beginthm|Theorem|(Vogt)}} If a 3-manifold $M$ admits a foliation by circles, then any 3-manifold obtained by removing finitely many points from $M$ admits a (not necessarily smooth) foliation by circles.{{endthm}} | ||
+ | |||
+ | {{beginthm|Corollary}} ${\mathbb R}^3$ admits a foliation by circles.{{endthm}} | ||
+ | </wikitex> | ||
== Further discussion == | == Further discussion == | ||
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{{#RefList:}} | {{#RefList:}} | ||
− | + | == External links == | |
+ | * The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Foliation foliations] | ||
+ | * The Wikipedia page about [[Wikipedia:Foliation|foliations]] | ||
[[Category:Manifolds]] | [[Category:Manifolds]] | ||
− |
Latest revision as of 13:07, 27 March 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
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 (smooth) foliation of
Tex syntax errorif
Tex syntax erroradmits an (smooth) atlas of foliated charts, that is (diffeomorphisms) homeomorphisms
The leaves of are the immersed 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
Tex syntax erroris given by .) A more general class are flat -bundles with or for a (smooth or topological) 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 . (
Tex syntax erroris a flat -bundle if .) Flat bundles fit into the frame work of fiber bundles. A (smooth) map
The 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
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 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 Taut foliations
Tex syntax erroris taut if for every leaf of there is a circle transverse to which intersects .
Theorem 2.1 (Rummler, Sullivan).
The following conditions are equivalent for transversely orientable codimension one foliations of closed, orientable, smooth manifoldsTex syntax error:
a) is taut;
b) there is a flow transverse to which preserves some volume form onTex syntax error; c) there is a Riemannian metric on
Tex syntax errorfor which the leaves of are least area surfaces.
2.6 Constructing new foliations from old ones
2.6.1 Pullbacks
[Candel&Conlon2000], Theorem 3.2.2
2.6.2 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 .
2.6.3 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 .
3 Invariants
3.1 Holonomy
Let be a foliation and a leaf. For a path contained in the intersection of the leaf with
a foliation chart , and two transversals to at the endpoints, the product structure of the foliation chart determines a homeomorphismThe composition yields a well-defined map from the germ of at to the germ of at , the so-called holonomy transport. The holonomy transport only depends on the relative homotopy class of .
Lemma 3.1.
Let be a foliation, a leaf, and a transversal at . Holonomy transport defines a homomorphismCorollary 3.2 (Reeb). Let be a transversely orientable codimension one foliation of a 3-manifold such that some leaf is homeomorphic to . Then and is the product foliation by spheres.
[Calegari2007] Theorem 4.5
3.2 Godbillon-Vey invariant
Tex syntax error, then its tangential plane field is defined by a nonsingular -form and for some . The Godbillon-Vey invariant of is defined as
The Godbillion-Vey invariant is related to resilience of leaves. A leaf is said to be resilient if it is not properly embedded and its holonomy is not trivial.
Theorem 3.3 (Duminy). If is a foliation of codimension one and no leaf is resilient, then .
4 Classification
4.1 Codimension one foliations
4.1.1 Existence
Theorem 4.1. 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.4.1.2 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 4.2.
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
4.1.3 Foliations of 3-manifolds
Tex syntax erroradmits a foliation without Reeb components, then , every leaf of is incompressible, and every transverse loop is essential in .
[Calegari2007] Theorem 4.37
A taut foliation has no Reeb component. IfTex syntax erroris an atoroidal 3-manifold, then, conversely, every foliation without Reeb components is taut.
Tex syntax errornot finitely covered by , then the universal covering
Tex syntax erroris homeomorphic to and the pull-back foliation is homeomorphic to a product foliation , where is a foliation of by lines.
[Calegari2007] Theorem 4.38
In particular, a taut foliation of a 3-manifoldTex syntax erroryields an action of on a (possibly non-Hausdorff) simply connected 1-manifold , the space of leaves of .
Tex syntax errorbe a closed, irreducible 3-manifold. a) If , then
Tex syntax erroradmits a taut foliation. b) If is a surface which minimizes the Thurston norm in its homology class , then
Tex syntax erroradmits a taut foliaton for which is a leaf.
4.2 Codimension two foliations
...
4.2.1 S1-foliations of 3-manifolds
Example 4.7. a) For every rational number there exists a foliaton of by circles such that restriction to the standard embedded torus is the suspension foliation of .
b) The complement of a knot admits a foliation by circles if and only if is a torus knot.Tex syntax erroradmits a foliation by circles, then any 3-manifold obtained by removing finitely many points from
Tex syntax erroradmits a (not necessarily smooth) foliation by circles.
5 Further discussion
...
6 References
- [Calegari2007] D. Calegari, Foliations and the geometry of 3-manifolds., Oxford Mathematical Monographs; Oxford Science Publications. Oxford University Press, Oxford, 2007. MR2327361 (2008k:57048) Zbl 1118.57002
- [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
7 External links
- The Encylopedia of Mathematics article on foliations
- The Wikipedia page about foliations