Poincaré - Reidemeister metric, Euler structures, and torsion

Details Poincaré - Reidemeister metric, Euler structures, and torsion Poincaré - Reidemeister metric, Euler structures, and torsion

1998-03-27

arxiv.org/abs/math/9803137v2

Mathematics

Differential Geometry

3 figures, AmsTex; Theorems 4.4 and 11.2 improved

Scientific paper

In this paper we define a Poincar\'e-Reidemeister scalar product on the determinant line of the cohomology of any flat vector bundle over a closed orientable odd-dimensional manifold. It is a combinatorial "torsion-type" invariant which refines the PR-metric, introduced earlier by the first author, and contains an additional sign or phase information. We compute the PR-scalar product in terms of the torsions of Euler structures, introduced earlier by the second author. We show that the sign of our PR-scalar product is determined by the Stiefel-Whitney classes and the semi-characteristic of the manifold. As an application, we compute the Ray-Singer analytic torsion via the torsions of Euler structures. Another application: a computation of the twisted semi-characteristic in terms of the Stiefel-Whitney classes.

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