Axioms for Higher Twisted Torsion Invariants of Smooth Bundles

TL;DR

This paper establishes axioms for twisted higher torsion invariants of smooth bundles, showing they are spanned by higher Franz Reidemeister torsion and twisted Miller-Morita-Mumford classes, with the dimension depending on the torsion degree.

Contribution

It introduces a system of axioms characterizing twisted higher torsion classes and identifies the span of these invariants in terms of known classes.

Findings

Twisted higher torsion invariants are either one or two dimensional.

These invariants are spanned by higher Franz Reidemeister torsion and twisted Miller-Morita-Mumford classes.

The dimension depends on the torsion degree.

Abstract

This paper attempts to investigate the space of various characteristic classes for smooth manifold bundles with local system on the total space inducing a finite holonomy covering. These classes are known as twisted higher torsion classes. We will give a system of axioms that we require these cohomology classes to satisfy. Higher Franz Reidemeister torsion and twisted versions of the higher Miller-Morita-Mumford classes will satisfy these axioms. We will show that the space oftwisted torsion invariants is two dimensional or one dimensional depending on the torsion degree and spanned by these two classes. The proof will greatly depend on results on the equivariant Hatcher constructions developed in a separate paper.