  A homology cylinder over a compact manifold is a homology cobordism between
two copies of the manifold together with a boundary parametrization. We study
abelian quotients of the homology cobordism group of homology cylinders. For
homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim
that their homology cobordism groups have infinitely generated abelian quotient
groups by using Reidemeister torsion invariants. In this paper, we first
investigate their abelian quotients again by using another invariant called the
Magnus representation. After that, we apply the machinery obtained from the
Magnus representation to higher dimensional cases and show that the homology
cobordism groups of homology cylinders over a certain series of manifolds
regarded as a generalization of surfaces have big abelian quotients. In the
proof, a homological localization, called the acyclic closure, of a free group
and its automorphism group play important roles and our result also provides
some information on these groups from a group-theoretical point of view.
