Relative torsion

Abstract

This paper achieves, among other things, the following:
• It frees the main result of [9] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations.
• It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [3] from finite dimensional representations of Γ to representations on an A-Hilbert module of finite type (A a finite von Neumann algebra). The result of [3] corresponds to A = C .
• It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M.
These objectives are achieved with the help of the relative torsion ℛ, first introduced by Carey, Mathai and Mishchenko [12] in special cases. The main result of this paper calculates explicitly this relative torsion (cf. Theorem 1.1).