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

Friedl, Stefan; Kim, Taehee

doi:10.1090/S0002-9947-08-04253-0

The parity of the Cochran–Harvey invariants of 3–manifolds
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by Stefan Friedl and Taehee Kim PDF

Trans. Amer. Math. Soc. 360 (2008), 2909-2922 Request permission

Abstract:

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.

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