Bernard Perron

Fox differential Calculus revisited.  Applications to mapping class group
of surfaces and low dimension topology.


Abstract: We begin by a review of Fox differential Calculus. Using this
Calculus we associate to any homeomorphism of surface a Fox matrix whose
entries belong to the group ring of the fundamental group of the surface.
From this we obtain a fair generalization of Johnson homomorphisms,
defined on various subgroups of the mapping class group. This allows us to
recover very easily deep results of Morita on the mapping class group.

Using the first part, we compute the Casson invariant of a 3-homology
sphere obtained by gluing two handlebodies along a homeomorphism of the
boundary belonging to the Torelli subgroup.

The third part (if we have time), still using Fox Calculus, will be on the
definition of a homotopy intersection theory on surfaces and punctured
2-disk, with applications to mapping class group and braid group.