6533b7d4fe1ef96bd1262e99

RESEARCH PRODUCT

Equivalence relations among homology 3-spheres and the Johnson filtration

subject

description

The Torelli group of a surface consists of isotopy classes of homeomorphisms of this surface acting trivially at the homological level. The structure of the Torelli group can be approached by the study and the comparison of two filtrations of this group: its lower central series, and the "Johnson" filtration, given by the kernels of the natural actions on the successive nilpotent quotients of the fundamental group of the surface. It is now known that there are, via the notion of "Heegaard splittings", rich interactions between this 2-dimensional study and the study of some 3-manifolds topological invariants: we refer here precisely to the so-called "finite-type" invariants. In this PhD, we are interested, through the study of the Torelli group, to some equivalence relations on homology 3-spheres. This allows us both to state results about homology 3-spheres and their surgeries, and results about the Johnson filtration of the Torelli group.Specifically, we study first the second Johnson homomorphism (a homomorphism defined on the second term of the Johnson filtration), and its interaction with the subgroup of elements extending to a handlebody bounded by the considered surface. This allows us to give new description of the set of homology 3-spheres. In a second part, we prove that a certain equivalence relation is trivial among homology 3-spheres. Two homology 3-spheres are always related by a surgery along an element of the fourth term of the Johnson filtration. This is shown by proving the surjectivity of the restriction of a homomorphism called "the core of the Casson invariant" to the fourth term of the Johnson filtration. In the proof, we exhibit a "non-trivial" element of the fourth term of the Johnson filtration.