Reidemeister–Turaev torsion modulo one of rational homology three-spheres

Given an oriented rational homology $3$ –sphere $M$ , it is known how to associate to any Spin $^{c}$ –structure $σ$ on $M$ two quadratic functions over the linking pairing. One quadratic function is derived from the reduction modulo $1$ of the Reidemeister–Turaev torsion of $(M, σ)$ , while the other one can be defined using the intersection pairing of an appropriate compact oriented $4$ –manifold with boundary $M$ . In this paper, using surgery presentations of the manifold $M$ , we prove that those two quadratic functions coincide. Our proof relies on the comparison between two distinct combinatorial descriptions of Spin $^{c}$ –structures on $M$ : Turaev’s charges vs Chern vectors.

Keywords

rational homology $3$–sphere, Reidemeister torsion, complex spin structure, quadratic function

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57Q10, 57R15

References

Forward citations

Publication

Received: 1 January 2003

Revised: 3 October 2003

Accepted: 7 November 2003

Published: 13 November 2003

Proposed: Robion Kirby

Seconded: Walter Neumann, Cameron Gordon

Authors

Florian Deloup
Laboratoire Emile Picard UMR 5580 CNRS/Univ. Paul Sabatier 118 route de Narbonne 31062 Toulouse Cedex 04 France

Gwenael Massuyeau
Laboratoire Jean Leray UMR 6629 CNRS/Univ. de Nantes 2 Rue de la Houssinière BP 92208 44322 Nantes Cedex 03 France

Volume 7, issue 2 (2003)

Florian Deloup and Gwenael Massuyeau