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

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.