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

Homology cylinders: an enlargement of the mapping class group