A remark on the higher torsion invariants for flat vector bundles with finite holonomy

TL;DR

This paper proves the equivalence of topological and analytic torsion invariants for flat vector bundles with finite rational holonomy, using representation theory and reducing to known trivial bundle cases.

Contribution

It extends the equivalence of torsion invariants to a broader class of flat bundles with finite rational holonomy, improving previous results by employing Artin's induction theorem.

Findings

Igusa-Klein topological torsion equals Bismut-Lott analytic torsion for specified bundles

Reduction to trivial bundle case via Artin's induction theorem

Improves upon previous work by Ohrt on the same topic

Abstract

We show that the Igusa-Klein topological torsion and the Bismut-Lott analytic torsion are equivalent for any flat vector bundle whose holonomy is a finite subgroup of $\mathrm{GL}_n(\mathbb{Q})$. Our proof uses Artin's induction theorem in representation theory to reduce the problem to the special case of trivial flat line bundles, which is a recent result of Puchol, Zhu and the second author. The idea of using Artin's induction theorem appeared in a paper of Ohrt on the same topic, of which our present work is an improvement.