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-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.

28 pages. The whole paper has been rewritten, and the title has been changed

References in corpus (3)

• Noncommutative knot theory
• Monotonicity of Degrees of Generalized Alexander Polynomials of Groups and 3-Manifolds
• Homology cylinders and the acyclic closure of a free group

Cited by in corpus (10)

• Homology cylinders and sutured manifolds for homologically fibered knots
• Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders
• The cobordism group of homology cylinders
• Three flavors of twisted invariants of knots
• Homology cylinders of higher-order
• Factorization formulas and computations of higher-order Alexander invariants for homologically fibered knots
• The Magnus representation and homology cobordism groups of homology cylinders
• Abelian quotients of monoids of homology cylinders
• Invariants and structures of the homology cobordism group of homology cylinders
• A non-commutative Reidemeister-Turaev torsion of homology cylinders