Filtrations on the monoid of homology cylinder and the Torelli group of surfaces

par Quentin Faes

mardi 24 mars 2026, 11:15 → 12:15 Europe/Paris

The mapping class group of a surface and its Torelli subgroup admit natural embeddings into monoids of homology cobordisms of 3-manifolds, providing a powerful bridge between 2-dimensional topology and 3-dimensional topology. In this talk, my main goal will be to show the audience how one can use this connection to obtain information about the algebraic structure of the Torelli group.

 

I will present results concerning the graded space associated with the so-called Y-filration of these monoids. This generalizes the lower central series of the Torelli group and is related to Goussarov and Habiro's clasper calculus, and thus to the theory of finite-type invariants of 3-dimensional manifolds. In particular we describe the second and third nilpotent quotients of the Torelli group, and we also discuss the existence of non-trivial torsion through elegant diagrammatic methods. (joint work with G. Massuyeau and M. Sato)