Comparison of Higher Smooth Torsion

TL;DR

This paper proves that two different constructions of higher torsion invariants for smooth bundles, one using Morse theory and the other using homotopy theory, are equivalent.

Contribution

It explicitly compares and confirms the equivalence of higher torsion invariants defined by different mathematical approaches.

Findings

Higher torsion invariants from Morse theory and homotopy theory are equivalent.

The comparison clarifies the relationship between different constructions of higher torsion.

The results unify different methods of defining higher torsion invariants.

Abstract

By explicitly comparing constructions, we prove that the higher torsion invariants of smooth bundles defined by Igusa and Klein via Morse theory agree with the higher torsion invariants defined by Badzioch, Dorabiala, Dwyer, Weiss, and Williams using homotopy theoretical methods.