Narrow Results

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.

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x^-1, y, y^-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C ⊗_L R〚x, y〛[(xy)^-1] and C ⊗_L R[x, x^-1]〚y〛[y^-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.