An $L^2$-Cheeger M\"uller theorem on compact manifolds-with-boundary

TL;DR

This paper extends the Cheeger-M"uller theorem to flat bundles on manifolds-with-boundary, incorporating recent anomaly results to handle more general boundary conditions.

Contribution

It generalizes the Cheeger-M"uller theorem to flat bundles with unimodular boundary restrictions on manifolds-with-boundary.

Findings

Proves an $L^2$-Cheeger-M"uller theorem for a broader class of flat bundles.

Utilizes recent anomaly results to handle boundary conditions.

Extends previous results to infinite covering spaces with boundary.

Abstract

We generalize a Cheeger-M\"uller type theorem for flat, unitary bundles on infinite covering spaces over manifolds-with-boundary, proven by Burghelea, Friedlander and Kappeller arXiv:dg-ga/9510010 [math.DG]. Employing recent anomaly results by Br\"uning, Ma and Zhang, we prove an analogous statement for a general flat bundle that is only required to have a unimodular restriction to the boundary.