Gwenael Massuyeau

Characterization of Y2-Equivalence for Homology Cylinders

For Σ a compact connected oriented surface, we consider homology cylinders over Σ: these are homology cobordisms with an extra homological triviality condition. When considered up to Y2-equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian group. In this paper, when Σ has one or zero boundary component, we define a surgery map from a certain space of graphs to this group. This map is shown to be an isomorphism, with inverse given by some extensions of the first Johnson homomorphism and Birman-Craggs homomorphisms.

Spin Borromean surgeries

In 1986, Matveev defined the notion of Borromean surgery for closed oriented 3-manifolds and showed that the equivalence relation generated by this move is characterized by the pair (first betti number, linking form up to isomorphism). We explain how this extends for 3-manifolds with spin structure if we replace the linking form by the quadratic form defined by the spin structure. We then show that the equivalence relation among closed spin 3-manifolds generated by spin Borromean surgeries is characterized by the triple (first betti number, linking form up to isomorphism, Rochlin invariant modulo 8).

Some finiteness properties for the Reidemeister-Turaev torsion of three-manifolds

We prove for the Reidemeister–Turaev torsion of closed oriented three-manifolds some finiteness properties in the sense of Goussarov and Habiro, that is, with respect to some cut-and-paste operations which preserve the homology type of the manifolds. In general, those properties require the manifolds to come equipped with an Euler structure and a homological parametrization.

Brackets in the Pontryagin Algebras of Manifolds

Given a smooth oriented manifold

Quadratic functions and complex spin structures on three-manifolds

M

Quadratic functions on torsion groups

with non-empty boundary, we study the Pontryagin algebra

Quadratic functions and complex spin structures on 3-manifolds

A=H_ast(Omega )

Brackets in loop algebras of manifolds

where

An introduction to the abelian Reidemeister torsion of three-dimensional manifolds

Omega

A functorial extension of the abelian Reidemeister torsions of three-manifolds

is the space of loops in