ar X iv : m at h / 98 03 13 7 v 2 [ m at h . D G ] 1 6 M ay 1 99 8 POINCARÉ - REIDEMEISTER METRIC , EULER STRUCTURES , AND TORSION
نویسنده
چکیده
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 in [Fa] and contains an additional sign or phase information. We compute the PR-scalar product in terms of the torsions of Euler structures, introduced in [T1], [T2]. 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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