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 and an epimorphism Cochran and Harvey defined a sequence of 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 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

Email:
friedl@math.rice.edu

**Taehee Kim**

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

Email:
tkim@konkuk.ac.kr

DOI:
https://doi.org/10.1090/S0002-9947-08-04253-0

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