  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-Wang's argument over the
Gassner representation of string links. Moreover, by applying Cochran and
Harvey's framework of higher-order (non-commutative) Alexander invariants to
them, we extract several pieces of information about the monoid and related
objects.
