## Tuesday Seminar on Topology

Seminar information archive ～03/29｜Next seminar｜Future seminars 03/30～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

### 2011/11/15

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Singular codimension-one foliations

and twisted open books in dimension 3.

(joint work with G. Meigniez)

(ENGLISH)

**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.

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.