The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces

Abstract

The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk, Livingston and Wang’s argument over the Gassner representation of string links. Then, by applying Cochran and Harvey’s framework of higher-order (noncommutative) Alexander invariants to them, we extract several information about the monoid and related objects.