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 (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.