The Parity of the Cochran–harvey Invariants of 3–manifolds

Given a finitely presented group G and an epimorphism φ : G → Z Cochran and Harvey defined a sequence of invariants dn(G,φ) ∈ N0, n ∈ N0, 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.