Foliations
Contents |
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 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
.
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 leaf space is (diffeomeorphic) homeomorphic to
.
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
.
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)
![\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 Constructing new foliations from old ones
2.5.1 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
2.5.2 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
![\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
.
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
![\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)
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.
![\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)
3.1.2 Foliations of surfaces
Codimension one foliations exist only on surfaces with
, 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![f:\rightarrow I](/images/math/f/7/e/f7e8a5687fdb7d63fd1a17432122ec16.png)
[Hector&Hirsch1981], Theorem 4.2.15 and Proposition 4.3.2
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
- [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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