The parity of the Cochran-Harvey invariants of 3-manifolds

Given a finitely presented group $G$ and an epimorphism $\phi:G\to \mathbb{Z}$ Cochran and Harvey defined a sequence of invariants $\overline{d}_n(G,\phi) \in \mathbb{N}_0, n\in \mathbb{N}_0$, which can be viewed as the degrees of higher-order Alexander polynomials. Cochran and Harvey showed that (up to a minor modification) this is a never decreasing sequence of numbers if $G$ is the fundamental group of a 3-manifold with empty or toroidal boundary. Furthermore they showed that these invariants give lower bounds on the Thurston norm.

Using a certain Cohn localization and the duality of Reidemeister torsion we show that for a fundamental group of a 3-manifold any jump in the sequence is necessarily even. This answers in particular a question of Cochran. Furthermore using results of Turaev we show that under a mild extra hypothesis the parity of the Cochran-Harvey invariant agrees with the parity of the Thurston norm.