A geometric description of the Reidemeister–Turaev torsion for abelian representations on 3-manifolds

Let M be a closed oriented 3-manifold equipped with an Euler structure e and an acyclic representation of its fundamental group. We define a twisted self-linking homology class of the diagonal of the two-point configuration space of M with respect to e. This twisted self-linking homology class appears as an obstruction in the Chern–Simons perturbation theory. When the representation is the maximal free abelian representation ρ₀, we prove that our self-linking class is a properly defined “logarithmic derivative” of the Reidemeister–Turaev torsion of (M, ρ₀, e) equipped with the given Euler structure.