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Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle. (English) Zbl 0830.58030
From the author’s abstract: “In this paper, we extend our previous results relating Milnor and Ray-Singer metrics on the determinant of the cohomology of a flat complex vector bundle to the equivariant case. Thus, we extend Lott and Rothenberg and Lück’s theorems relating equivariant combinatorial and analytic torsions to flat vector bundles which are not necessarily unitary flat.”
The main result extends to the equivariant case, Theorem 0.2 of the authors’ monograph [An extension of a theorem by Cheeger and Müller, Astérisque 205 (1992; Zbl 0781.58039)]. The authors also show, how to avoid using the difficult Helffer-Sjöstrand results, that derive essential properties of the Thom-Smale complex from the analysis of the Witten complex. Instead, they use known properties of the Thom-Smale complex, including those obtained by Laudenbach in the Appendix to the authors paper [loc. cit.] to get the information they need on the Witten complex, even recovering refined results of Helffer-Sjöstrand on the asymptotics of the Witten complex.

MSC:
58J52 Determinants and determinant bundles, analytic torsion
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References:
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