Novikov homology and non-commutative Alexander polynomials

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.