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.
"Homology cylinders: an enlargement of the mapping class group." Algebr. Geom. Topol. 1 (1) 243 - 270, 2001. https://doi.org/10.2140/agt.2001.1.243