  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.
