Intersection torsion and analytic torsion of spaces with conical singularities

TL;DR

This paper extends the Cheeger-Müller theorem to spaces with isolated conical singularities, showing the equality of analytic and intersection torsion in even dimensions and analyzing anomalies in odd dimensions.

Contribution

It develops a combinatorial intersection homology theory and spectral analysis for the Hodge-Laplace operator on conical singular spaces, extending classical torsion results.

Findings

Analytic torsion coincides with intersection torsion in even dimensions.

The ratio of torsions is non-trivial in odd dimensions, depending on the link of singularities.

Extension of Hodge theory and torsion results to conical singular spaces.

Abstract

We prove an extension of the Cheeger-M\"{u}ller theorem to spaces with isolated conical singularities: the $L^2$-analytic torsion coincides with the Ray-Singer intersection torsion on an even dimensional space, and they are trivial, while the ratio is non trivial on an odd dimensional space, and the anomaly depends only on the link of the singularities. For this aim, we develop on one side a combinatorial cellular theory whose homology coincides with the intersection homology of Gregory and Macpherson, and where the Ray-Singer intersection torsion is well defined. On the other side, we elaborate the spectral theory for the Hodge-Laplace operator on the square integrable forms on a space with conical singularities {\it \'a la} Cheeger, and we extend the classical results of the Hodge theory and the analytic torsion.