Non-commutative Reidemeister torsion and Morse-Novikov theory

TL;DR

This paper generalizes Reidemeister torsion to non-commutative settings for circle-valued Morse functions, linking it to Novikov theory and providing a Morse-theoretic and dynamical interpretation.

Contribution

It extends Hutchings and Lee's abelian coefficient results to non-commutative skew fields, connecting torsion, zeta functions, and Novikov complexes.

Findings

Reidemeister torsion equals the product of a non-commutative zeta function and algebraic torsion.

Provides a Morse-theoretic and dynamical interpretation of higher-order Reidemeister torsion.

Generalizes previous abelian coefficient results to non-commutative rings.

Abstract

Given a circle-valued Morse function of a closed oriented manifold, we prove that Reidemeister torsion over a non-commutative formal Laurent polynomial ring equals the product of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of the Novikov complex over the ring. This paper gives a generalization of the result of Hutchings and Lee on abelian coefficients to the case of skew fields. As a consequence we obtain a Morse theoretical and dynamical description of the higher-order Reidemeister torsion.