#### Volume 7, issue 2 (2003)

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

### Florian Deloup and Gwenael Massuyeau

Geometry & Topology 7 (2003) 773–787
 arXiv: math.GT/0301041
##### 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 $\left(M,\sigma \right)$, 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.

##### Keywords
rational homology $3$–sphere, Reidemeister torsion, complex spin structure, quadratic function
##### Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57Q10, 57R15