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Abelian quotients of monoids of homology cylinders

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.