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Poincaré - Reidemeister metric, Euler structures, and torsion

Michael Farber·Vladimir Turaev
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Abstract

In this paper we define a Poincaré-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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Michael Farber, Vladimir Turaev. 1998-05-16. Poincaré - Reidemeister metric, Euler structures, and torsion. https://arxiv.org/abs/math/9803137

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