Foliations
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1 Introduction
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
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 Taut foliations
A codimension one foliation of is 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 manifolds :
a) is taut;
b) there is a flow transverse to which preserves some volume form on ;
c) there is a Riemannian metric on for 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
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
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
[Calegari2007] Theorem 4.37
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.
[Calegari2007] Theorem 4.38
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 .
Theorem 4.5 (Gabai). Let be a closed, irreducible 3-manifold.
a) If , then admits a taut foliation.
b) If is a surface which minimizes the Thurston norm in its homology class , then admits a taut foliaton for which is a leaf.1 Codimension two foliations
1.1 -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.2 Further discussion
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5 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
This page has not been refereed. The information given here might be incomplete or provisional. |