The parity of the Cochran-Harvey invariants of 3-manifolds
Abstract
Description
Given a finitely presented group G and an epimorphism G to the group of integers Cochran and Harvey defined a sequence of integral invariants, 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, and 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.
15 pages, accepted for publicaton by the Transactions of the AMS
15 pages, accepted for publicaton by the Transactions of the AMS