Torus bundles and the group cohomology of GL(N,Z)

TL;DR

This paper proves the vanishing of a specific characteristic class for flat vector bundles with structure groups in GL(N,Z) by explicitly expressing it as an exact form, advancing understanding of flat bundle invariants.

Contribution

It provides an explicit construction showing the characteristic class vanishes for bundles with structure group in GL(N,Z), a novel result in the study of flat vector bundles.

Findings

Characteristic class vanishes for bundles with structure group in GL(N,Z)

Explicit expression of the characteristic class as an exact form

Advances understanding of flat bundle invariants

Abstract

We prove the vanishing of a certain characteristic class of flat vector bundles when the structure groups of the bundles are contained in GL(N,Z). We do so by explicitly writing the characteristic class as an exact form on the base of the bundle.