Algebraic and Geometric Topology 2 (2002), paper no. 7, pages 137-155.

A norm for the cohomology of 2-complexes

Vladimir Turaev

Abstract. We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-complex X, the norm on H^1(X; R) determined by graphs on X majorates the Alexander-Fox norms derived from \pi_1(X).

Keywords. Group cohomology, norms, 2-complexes, Alexander-Fox polynomials

AMS subject classification. Primary: 57M20. Secondary: 57M05.

DOI: 10.2140/agt.2002.2.137

E-print: arXiv:math.AT/0203042

Submitted: 1 October 2001. Accepted: 6 February 2002. Published: 28 February 2002.

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Vladimir Turaev
IRMA, Universite Louis Pasteur -- CNRS
7 rue Rene Descartes, 67084 Strasbourg, France

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