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

Sakasai, Takuya

doi:10.2140/agt.2008.8.803

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

Takuya Sakasai

Algebraic & Geometric Topology 8 (2008) 803–848

DOI: 10.2140/agt.2008.8.803

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.

Keywords

homology cylinder, Magnus representation, higher-order Alexander invariant, string link, Reidemeister torsion, Dieudonné determinant

Mathematical Subject Classification 2000

Primary: 57M05

Secondary: 57M27, 20F34, 57N05

References

Publication

Received: 30 November 2006

Accepted: 23 January 2008

Published: 3 June 2008

Authors

Takuya Sakasai
Graduate School of Mathematical Sciences The University of Tokyo 3-8-1 Komaba, Meguro, Tokyo 153-8914 Japan
http://www.ms.u-tokyo.ac.jp/~sakasai/

Volume 8, issue 2 (2008)