Laboratoire de Mathématiques Jean Leray

Le but de cet exposé est de montrer comment les techniques classiques de l'algèbre homotopique permettent de revisiter la théorie de la cohomologie d'intersection de Goresky-McPherson. En particulier, on montre comment définir un cup-produit d'intersection globalement défini au niveau des cochaines. Il s'agit d'un travail en cours avec Martin Saralegi et Daniel Tanré.

Given a result of M. Herman, we provide a new elementary proof of the fact that the connected component of the group of compactly supported diffeomorphisms is perfect and hence simple. Moreover, we show that every diffeomorphism g, which is sufficiently close to the identity, can be represented as a product of four commutators, g = [h_1 , k_1 ]· · ·[h4 , k4 ], where the factors h_i and k_i can be chosen to depend smoothly on g. Next the obtained results will be compared with classical and recent theorems on the structure of diffeomorphism groups.

This talk is for the International Week celebrating the 50th anniversary of the University of Nantes. Any knot can be regarded as the boundary of some curved surface. The curved surface is called a Seifert surface. I introduce an algorithm to obtain a Seifert surface from a knot. By using Seifert surfaces and Morse functions, we can define an invariant for knots. We discuss some recent topics on the invariant.