On the Cheeger-Müller theorem for an even-dimensional cone (Q2786974)

The Cheeger-Müller theorem states the equality of analytic and combinatorial torsions of a closed manifold with coefficients in a flat orthogonal bundle. It has been extended to manifolds with boundary, in various degrees of generality, by \textit{W. Lück} [J. Differ. Geom. 37, No. 2, 263--322 (1993; Zbl 0792.53025)], \textit{S. M. Vishik} [Commun. Math. Phys. 167, No. 1, 1--102 (1995; Zbl 0818.58049)] and \textit{J. Brüning} and \textit{X. Ma} [Math. Z. 273, No. 3--4, 1085--1117 (2013; Zbl 1318.58018); erratum ibid. 278, No. 1--2, 615--616 (2014)]: in this case it is not an equality but a geometric expression for the difference between the torsions which depend only on the geometry of the boundary. In the present paper, the authors present a statement for metric cones over closed manifolds. It is hopefully a first step towards a Cheeger-Müller theorem for manifolds with isolated conical singularities, or even for more general stratified spaces.NEWLINENEWLINEIn the setting of stratified spaces it is not in fact obvious to define what should be the combinatorial torsion under consideration. In this paper, the authors use the intersection torsion defined by \textit{A. Dar} [Math. Z. 194, 193--216 (1987; Zbl 0605.57012)] using the intersection homology of Goresky-McPherson and they obtain an error term which is the same as for the smooth case. The proofs are by computing explicitly both torsions in terms of the boundary (the basis of the cone). The details for the analytic side are given in another paper by the same authors [J. Geom. Phys. 61, No. 3, 624--657 (2011; Zbl 1222.58027)].