Novikov homology and noncommutative Alexander polynomials

TL;DR

This paper explores the connection between Novikov-Sikorav homology and non-commutative Alexander polynomials, proposing a new perspective on their role in understanding 3-manifold topology and Thurston norm bounds.

Contribution

It introduces the idea that the vanishing of a specific Novikov-Sikorav homology module defines a monic non-commutative Alexander polynomial, providing new proofs in 3D topology.

Findings

Vanishing of Novikov-Sikorav homology characterizes monic non-commutative Alexander polynomials.

New proofs of properties of Novikov-Sikorav homology in 3-manifold topology.

Establishes a link between homology vanishing and Alexander polynomial monicness.

Abstract

In the early 2000's Cochran and Harvey introduced non-commutative Alexander polynomials for 3-manifolds. Their degrees give strong lower bounds on the Thurston norm. In this paper we make the case that the vanishing of a certain Novikov-Sikorav homology module is the correct notion of a monic non-commutative Alexander polynomial. Furthermore we will use the opportunity to give new proofs of several statements about Novikov-Sikorav homology in the three-dimensional context.