Combinatorial invariants computing the Ray-Singer analytic torsion

Abstract

It is shown that for any piecewise-linear closed orientable manifold of odd dimension there exists an invariantly defined metric on the determinant line of cohomology with coefficients in an arbitrary flat bundle E over the manifold (E is not required to be unimodular). The construction of this metric (called Poincare - Reidemeister metric) is purely combinatorial; it combines the standard Reidemeister type construction with Poincare duality. The main result of the paper states that the Poincare-Reidemeister metric computes combinatorially the Ray-Singer metric. It is shown also that the Ray-Singer metrics on some relative determinant lines can be computed combinatorially (including the even-dimensional case) in terms of metrics determined by correspondences.