## Tuesday Seminar on Topology

Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya

### 2011/11/15

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Francois Laudenbach (Univ. de Nantes)
Singular codimension-one foliations
and twisted open books in dimension 3.
(joint work with G. Meigniez)
(ENGLISH)
[ Abstract ]
The allowed singularities are those of functions.
According to A. Haefliger (1958),
such structures on manifolds, called $\\Gamma_1$-structures,
are objects of a cohomological
theory with a classifying space $B\\Gamma_1$.
The problem of cancelling the singularities
(or regularization problem)
arise naturally.
For a closed manifold, it was solved by W.Thurston in a famous paper
(1976), with a proof relying on Mather's isomorphism (1971):
Diff$^\\infty(\\mathbb R)$ as a discrete group has the same homology
as the based loop space
$\\Omega B\\Gamma_1^+$.
For further extension to contact geometry, it is necessary
to solve the regularization problem
without using Mather's isomorphism.
That is what we have done in dimension 3. Our result is the following.

{\\it Every $\\Gamma_1$-structure $\\xi$ on a 3-manifold $M$ whose
normal bundle
embeds into the tangent bundle to $M$ is $\\Gamma_1$-homotopic
to a regular foliation
carried by a (possibily twisted) open book.}

The proof is elementary and relies on the dynamics of a (twisted)
pseudo-gradient of $\\xi$.

All the objects will be defined in the talk, in particular the notion
of twisted open book which is a central object in the reported paper.