The Magnus representation and homology cobordism groups of homology cylinders

Abstract

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.