Homology cylinders: an enlargement of the mapping class group

Abstract

We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov–Habiro theory of finite-type invariants of 3–manifolds. We study the structure of this group from several directions: the relative weight filtration of Dennis Johnson, the finite-type filtration of Goussarov–Habiro, and the relation to string link concordance.

We also consider a new Lagrangian filtration of both the mapping class group and the group of homology cylinders.