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

Authors: Stefan Friedl and Taehee Kim
Journal: Trans. Amer. Math. Soc. 360 (2008), 2909-2922
MSC (2000): Primary 57M27; Secondary 57M05
Published electronically: January 7, 2008
MathSciNet review: 2379780
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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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Additional Information

Stefan Friedl
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005-1892
Address at time of publication: Département de Mathématiques, UQAM, C.P. 8888, Succursale Centre-ville, Montréal, Quebec, Canada H3C 3P8

Taehee Kim
Affiliation: Department of Mathematics, Konkuk University, Hwayang-dong, Gwangjin-gu, Seoul 143-701, Korea

Keywords: Thurston norm, 3-manifold groups, Alexander polynomials
Received by editor(s): October 25, 2005
Received by editor(s) in revised form: February 13, 2006
Published electronically: January 7, 2008
Additional Notes: The second author is the corresponding author for this paper
Article copyright: © Copyright 2008 American Mathematical Society

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