Combinatorial Invariants Computing the Ray-singer Analytic Torsion

Let K denote a closed odd-dimensional smooth manifold and let E be a flat vector bundle over K. In this situation the construction of Ray and Singer [RS] gives a metric on the determinant line of the cohomology detH(M ;E) which is a smooth invariant of the manifold M and the flat bundle E. (Note that if the dimension of K is even then the Ray-Singer metric depends on the choice of a Riemannian metric on K and of a Hermitian metric on E). The famous theorem which was proved by J. Cheeger [C] and W.Müller [Mu], states that assuming that the flat vector bundle E is unitary (i.e. E admits a flat Hermitian metric), the Ray-Singer metric coincides with the Reidemeister metric which is defined using finite dimensional linear algebra by the combinatorial structure of K. The construction of the Reidemeister metric works also under a weeker assumption that the flat bundle E is unimodular, i.e. the line bundle det(E) admits a flat metric. In a recent paper W.Müller [Mu1] proved that in this case the Ray-Singer metric again coincides with the Reidemeister metric. Without the assumption that the flat bundle E is unimodular, the standard construction of Reidemeister metric is ambiguous (it depends on different choices made in the process of its construction). In this general situation J.-M. Bismut and W. Zhang [BZ] computed the deviation of the Ray-Singer metric from so called Milnor metric, cf. [BZ]; the result of their computation is given in a form of an integral of a Chern-Simons current which compensates the ambiguity of the Milnor metric and depends on the Riemannian metric on K and on the metric on E. In this paper I show 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. The construction of this metric is purely combinatorial. I call this metric Poincaré Reidemeister metric since it is defined by combining the standard Reidemeister’s construction with the Poincaré duality. The idea to use the Poincaré duality in order to construct an invariantly defined metric on the determinant line was prompted by the theorem 4.1 of D.Burghelea, L.Friedlander, and T.Kappeler [BFK], expressing the analytic torsion through some