Axioms for higher twisted torsion invariants of smooth bundles

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 of twisted torsion invariants is two-dimensional or one-dimensional depending on the torsion degree and is spanned by these two classes. The proof will greatly depend on results about the equivariant Hatcher constructions developed in Goodwillie, Igusa and Ohrt (2015).