Foliations
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1 Introduction
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![{\mathcal{F}}=\left\{F_b\right\}_{b\in B}](/images/math/5/8/2/582ce2c234750bdb9e45992b4d40168e.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![n-q](/images/math/0/f/8/0f87ac864ce49eabb7c316bcd1406f6d.png)
![{\mathcal{F}}](/images/math/c/e/3/ce3e88fa7de4d07aab00d63c4fa92550.png)
![q](/images/math/e/b/6/eb6af5b4e510c3c874d7d1f51d72393a.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}](/images/math/4/9/9/499a95bc8f177e4128385f3e481feeb2.png)
![\displaystyle \phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q](/images/math/d/8/f/d8fdbbc43ea0e49f673ee10c19748cee.png)
![\alpha\in{\mathcal{A}}, b\in B](/images/math/6/4/a/64a4ddcabd4b031c0c94e78b2135bb26.png)
![F_b\cap U_\alpha](/images/math/8/c/e/8ce7ce6a751e163e73835dd2fb1f66c9.png)
![\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)](/images/math/e/6/9/e692810a769ace8d5d09c08eebf0564d.png)
The leaves of are the immersed submanifolds
. Each
belongs to a unique leaf. The foliation
determines its tangential plane field
by
if
.
![\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}](/images/math/4/4/c/44cb27120af339f4425baff3db7926fa.png)
![\displaystyle \gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).](/images/math/c/0/8/c082ba44d4825a554b5680877e304a6a.png)
A 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
,
![\displaystyle \omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.](/images/math/a/7/f/a7f526cba769514962f9f40bc48ea7a1.png)
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
.)
![\displaystyle \pi:M\rightarrow B](/images/math/f/0/b/f0b4cb73c50ff2f4b0f6f2fb3e76d1e8.png)
![b\in B](/images/math/2/5/3/253abda89bc461fcefc63b886908df9b.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![\phi:\pi^{-1}\left(U\right)\rightarrow U\times F](/images/math/a/a/3/aa3c12d737abf9d9a276282d1f8cbe33.png)
![p_1](/images/math/d/f/4/df47f38ae6dff857e284de5a2b0a719d.png)
![\displaystyle \begin{xy} \xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\ U\ar[r]^{id}&U} \end{xy}](/images/math/7/2/5/72500c17d10d6bf99f9c071ac752f9d6.png)
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
![\displaystyle L_f:= \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\mbox{\ for\ }f\in F,](/images/math/f/0/9/f09c8e83ad49f44815cb7ea502a714ac.png)
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
![\displaystyle f:M\rightarrow B](/images/math/d/1/e/d1ec86ebf93882249e1bf711ab1c3067.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\pi^{-1}\left(b\right), b\in B](/images/math/f/4/1/f41423df7aebca672cf1a008efd345c0.png)
An example of a submersion, which is not a fiber bundle, is given by
![\displaystyle f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}](/images/math/0/5/2/052fdc10e3d29facd86cb6b6e41e53de.png)
![\displaystyle f\left(x,y\right)=\left(x^2-1\right)e^y.](/images/math/c/7/9/c798943883a02d60c214c93ddd9f0ed6.png)
![\left[-1,1\right]\times{\mathbb R}](/images/math/5/e/0/5e00477f106f9f9bfc1dcd3bfb0c4693.png)
![{\mathbb Z}](/images/math/3/6/6/3668f46c780d9ac8f67fd1ace5246dd5.png)
![\displaystyle z\left(x,y\right)=\left(x,y+z\right)](/images/math/7/7/5/775e4638d8317694910bc334ba8fc9b9.png)
![\displaystyle z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)](/images/math/a/7/c/a7ca8fd235922fac53bbf60cffdc6bc7.png)
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
![\displaystyle f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}](/images/math/7/0/1/7013c07bbd1d1b2f4ff07ebf4d65fb1b.png)
![\displaystyle f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,](/images/math/a/8/b/a8bdf10053c5ef102036e61a8cd58481.png)
![\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}](/images/math/3/c/b/3cb41268dbbd7e42abfb3366f15c615a.png)
![D^n](/images/math/f/8/3/f833fbe39d3dba9849c0729b3ce0a7f8.png)
![D^n\times{\mathbb R}](/images/math/d/8/6/d867a2fb5ea93c4fab81dceae039e99b.png)
![{\mathbb Z}](/images/math/3/6/6/3668f46c780d9ac8f67fd1ace5246dd5.png)
![\displaystyle z\left(x,y\right)=\left(x,y+z\right)](/images/math/7/7/5/775e4638d8317694910bc334ba8fc9b9.png)
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
;
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\mathcal{F}](/images/math/4/a/e/4ae36a11c000df5e78ad81c8f004d706.png)
2.6 Constructing new foliations from old ones
2.6.1 Pullbacks
![\left(M,{\mathcal{F}}\right)](/images/math/b/8/a/b8ab753294f8a1174a728a35079255c0.png)
![q](/images/math/e/b/6/eb6af5b4e510c3c874d7d1f51d72393a.png)
![f:N\rightarrow M](/images/math/c/c/4/cc42a2db5fdff05de37bec8ce6fb0c82.png)
![\mathcal{F}](/images/math/4/a/e/4ae36a11c000df5e78ad81c8f004d706.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f^{-1}\left(L\right)](/images/math/b/1/c/b1cdfe0e6db3ab75e8f716400531be13.png)
![L](/images/math/9/3/4/934cf4f6e8d65e941a602d24451533b6.png)
![\mathcal{F}](/images/math/4/a/e/4ae36a11c000df5e78ad81c8f004d706.png)
[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
![\displaystyle cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,](/images/math/b/8/3/b8379c6a609d77ea62536e3911cd00d0.png)
![\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}](/images/math/3/f/b/3fbf98245ae610938482eae5495cd808.png)
![\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]](/images/math/8/c/b/8cb553a8127f8d2b0f60d91295f357f5.png)
![\displaystyle \lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.](/images/math/a/1/d/a1ddc238831322e1d4b47eccae92edd7.png)
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
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![\tau_0,\tau_1](/images/math/f/f/3/ff32b021b8ee97de4b6caa13c44f3819.png)
![\gamma](/images/math/3/a/1/3a1eda9790fa00ea300950605481fc0d.png)
![\displaystyle h:\tau_0\mid_U\rightarrow \tau_1\mid_U.](/images/math/5/b/0/5b08e3c759a752023b4994189953752e.png)
![\gamma](/images/math/3/a/1/3a1eda9790fa00ea300950605481fc0d.png)
![U_0,\ldots,U_k](/images/math/7/6/a/76a9478d24ab4fe9052b341d3a7b6ba3.png)
![\displaystyle h_0:\tau_0\mid_{U_0}\rightarrow \tau_1\mid_{U_0},\ldots,h_k:\tau_k\mid_{U_k}\rightarrow \tau_{k+1}\mid_{U_{k+1}}.](/images/math/4/1/6/416d72ef5e83cd0d8516617a4b5dbae7.png)
The 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![\left(M,{\mathcal{F}}\right)](/images/math/b/8/a/b8ab753294f8a1174a728a35079255c0.png)
![L](/images/math/9/3/4/934cf4f6e8d65e941a602d24451533b6.png)
![x\in L](/images/math/a/6/b/a6b3e59126d2997fb11f9808ac4e22bd.png)
![\tau](/images/math/2/4/f/24f649f2eaad83d8a6a97f8e49fc6fac.png)
![x](/images/math/8/7/2/8725029ea89712eed8670bae64d30e47.png)
![\displaystyle H:\pi_1\left(L,x\right)\rightarrow {\mathcal{H}}omeo\left(\tau\right)](/images/math/5/5/e/55e532121e57e757caad1b402564df1d.png)
![\tau](/images/math/2/4/f/24f649f2eaad83d8a6a97f8e49fc6fac.png)
Corollary 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
![\displaystyle gv\left({\mathcal{F}}\right):=\left[\eta\wedge\left(d\eta\right)^q\right]\in H^{2q+1}_{dR}\left(M\right).](/images/math/3/b/2/3b2c7cfde68606b7d423ffdf55998ae0.png)
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.
![\chi(M^n)=0](/images/math/3/f/9/3f9bb5f110d02b57e2701cce657f36bc.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![\tau^{n-1}](/images/math/b/6/d/b6d01e855196a8100d43df83505f48dd.png)
![M^n](/images/math/b/5/d/b5da61f940225ab4af35bb26e7339b1a.png)
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![f:I\rightarrow I](/images/math/f/c/7/fc70d1f16eff5647d7225bfacfb4e8b9.png)
[Hector&Hirsch1981], Theorem 4.2.15 and Proposition 4.3.2
4.1.3 Foliations of 3-manifolds
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\mathcal{F}](/images/math/4/a/e/4ae36a11c000df5e78ad81c8f004d706.png)
![\pi_2\left(M\right)=0](/images/math/b/c/7/bc75863b290bfc7579c259bd29802222.png)
![\mathcal{F}](/images/math/4/a/e/4ae36a11c000df5e78ad81c8f004d706.png)
![\pi_1\left(M\right)](/images/math/9/5/8/958482685cf6f2b6fcfe25bf36b26ed0.png)
[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.
![\mathcal{F}](/images/math/4/a/e/4ae36a11c000df5e78ad81c8f004d706.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![S^2\times S^1](/images/math/5/1/7/517bb87a54a2217da634d98cc79a62f0.png)
![\widetilde{M}](/images/math/a/c/4/ac49ca39301be0e9cc576dc7efc5f6e6.png)
![{\mathbb R}^3](/images/math/a/6/b/a6b25d052fa9007721b022fc8400ef80.png)
![\left(\widetilde{M},\widetilde{\mathcal{F}}\right)](/images/math/c/8/a/c8a2c84fba4bb6e7c1b6e5d2493249b1.png)
![\left({\mathbb R}^2,{\mathcal{G}}\right)\times{\Bbb R}](/images/math/a/f/0/af06a64ae6f6d6c761c1297a46b33c01.png)
![\mathcal{G}](/images/math/5/1/7/517d0e03d4295fddc9e0e75328f335be.png)
![{\mathbb R}^2](/images/math/0/f/5/0f59dad65564793772c64a39a395d36f.png)
[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.
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![\left[S\right]\in H_2\left(M;{\mathbb R}\right)](/images/math/8/7/5/875ecfe79ac320198542724d4cacb126.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
1 Codimension two foliations
1.1
-foliations of 3-manifolds
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![{\mathbb R}^3](/images/math/a/6/b/a6b25d052fa9007721b022fc8400ef80.png)
2 Further discussion
...
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. |