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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces

Takuya Sakasai

Algebraic & Geometric Topology 8 (2008) 803–848

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.

homology cylinder, Magnus representation, higher-order Alexander invariant, string link, Reidemeister torsion, Dieudonné determinant
Mathematical Subject Classification 2000
Primary: 57M05
Secondary: 57M27, 20F34, 57N05
Received: 30 November 2006
Accepted: 23 January 2008
Published: 3 June 2008
Takuya Sakasai
Graduate School of Mathematical Sciences
The University of Tokyo
3-8-1 Komaba, Meguro, Tokyo