Relative torsion

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