The parity of the Cochran–Harvey invariants of 3–manifolds
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- by Stefan Friedl and Taehee Kim PDF
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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.References
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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
- MR Author ID: 746949
- Email: friedl@math.rice.edu
- Taehee Kim
- Affiliation: Department of Mathematics, Konkuk University, Hwayang-dong, Gwangjin-gu, Seoul 143-701, Korea
- MR Author ID: 743933
- Email: tkim@konkuk.ac.kr
- 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
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 2909-2922
- MSC (2000): Primary 57M27; Secondary 57M05
- DOI: https://doi.org/10.1090/S0002-9947-08-04253-0
- MathSciNet review: 2379780