Geometry & Topology, Vol. 7 (2003) Paper no. 22, pages 773--787.

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

Florian Deloup and Gwenael Massuyeau

Abstract. Given an oriented rational homology 3-sphere M, it is known how to associate to any Spin^c-structure \sigma 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,\sigma ), 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

AMS subject classification. Primary: 57M27. Secondary: 57Q10, 57R15.

DOI: 10.2140/gt.2003.7.773

E-print: arXiv:math.GT/0301041

Submitted to GT on 1 January 2003. (Revised 3 October 2003.) Paper accepted 7 November 2003. Paper published 13 November 2003.

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Florian Deloup, Gwenael Massuyeau
Laboratoire Emile Picard, UMR 5580 CNRS/Univ. Paul Sabatier
118 route de Narbonne, 31062 Toulouse Cedex 04, France
and
Laboratoire Jean Leray, UMR 6629 CNRS/Univ. de Nantes
2 rue de la Houssiniere, BP 92208, 44322 Nantes Cedex 03, France

Email: deloup@picard.ups-tlse.fr, massuyea@math.univ-nantes.fr

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