| Résume | Redemeister–Singer theorem states that, up to homeomorphism, any
compact connected oriented 3-manifold can be obtained by gluing two
handlebodies together. This connects the study of 3-manifolds to the
study of the mapping class group of surfaces. For instance, one can get
all homology 3-spheres by restricting the gluing map to be an element
acting trivially on the homology of the surface, i.e. an element of the
Torelli group. Another point of view is to say that one can get any
homology 3-sphere from 𝕊3 by performing the following surgery :
remove a handlebody and glue it back with an element of the Torelli
group. Somewhat surprisingly, we shall prove in this talk that we can
actually suppose this surgery to be performed with an element of the
4-th term of the Johnson filtration, i.e. an element acting trivially on
the 4-th nilpotent quotient of the fundamental group of the surface.
This result is an improvement of results obtained successively by Morita
and Pitsch. It is obtained by using Goussarov–Habiro clasper calculus, a
formula by Morita computing the Casson invariant of homology 3-spheres,
and a formula by Kawazumi and Kuno that encodes the action of a Dehn
twist on the fundamental group. |
