  A homology cylinder over a surface consists of a homology cobordism between
two copies of the surface and markings of its boundary. The set of isomorphism
classes of homology cylinders over a fixed surface has a natural monoid
structure and it is known that this monoid can be seen as an enlargement of the
mapping class group of the surface. We now focus on abelian quotients of this
monoid. We show that both the monoid of all homology cylinders and that of
irreducible homology cylinders are not finitely generated and moreover they
have big abelian quotients. These properties contrast with the fact that the
mapping class group is perfect in general. The proof is given by applying
sutured Floer homology theory to homologically fibered knots studied in a
previous paper.
