Combinatorial invariants computing the Ray-Singer analytic torsion

Abstract

It is shown that for any piecewise-linear closed orientable manifold K of odd dimension there exits an invariantly defined metric on the determinant line of cohomology det( H ∗(K; E) ), where E is an arbitrary flat bundle over K (here E is not required to be unimodular). The construction of this metric (called Poincaré-Reidemeister metric) is purely combinatorial; it combines the standard Reidemeister's construction with the Poincaré duality. The main properties of the Poincaré-Reidemeister metric consist in the following: (a) the Poincaré-Reidemeister metric can be computed starting from any polyhedral cell decomposition of the manifold in purely combinatorial terms; (b) the Poincaré-Reidemeister metric coincides with the Reidemeister metric when the latter is correctly defined (i.e., when the bundle E is unitary or unimodular). (c) The construction of Ray and Singer, which uses zeta-function regularized determinants of Laplacians, produces the metric on the determinant of cohomology, which coincides (via the De Rham isomorphisms) with the Poincaré-Reidemeister metric. This is the main result of the paper, showing that the Poincaré-Reidemeister metric computes combinatorially the Ray-Singer metric. (d) The Poincaré-Reidemeister metric behaves well with respect to natural correspondences between determinant lines which are discussed in the paper. 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 the metrics determined by correspondences.