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

Deloup, Florian¹ ; Massuyeau, Gwenael²

¹ Laboratoire Emile Picard, UMR 5580 CNRS/Univ. Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France
² Laboratoire Jean Leray, UMR 6629 CNRS/Univ. de Nantes, 2 Rue de la Houssinière, BP 92208, 44322 Nantes Cedex 03, France

Geometry & topology, Tome 7 (2003) no. 2, pp. 773-787.

Résumé

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 $\left(M,\sigma \right)$, 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.

Détail

DOI : 10.2140/gt.2003.7.773

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

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