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The stable braid group and the determinant of the Burau representation

F R Cohen and J Pakianathan

Geometry & Topology Monographs 10 (2007) 117–129

DOI: 10.2140/gtm.2007.10.117

arXiv: math.AT/0509577

Abstract

This article gives certain fibre bundles associated to the braid groups which are obtained from a translation as well as conjugation on the complex plane. The local coefficient systems on the level of homology for these bundles are given in terms of the determinant of the Burau representation.

De Concini, Procesi, and Salvetti [Topology 40 (2001) 739–751] considered the cohomology of the nth braid group Bn with local coefficients obtained from the determinant of the Burau representation, H*(Bn;Q[t±1]). They show that these cohomology groups are given in terms of cyclotomic fields.

This article gives the homology of the stable braid group with local coefficients obtained from the determinant of the Burau representation. The main result is an isomorphism H*(B;F[t±1])→H*2S3⟨3⟩;F) for any field F where Ω2S3⟨3⟩ denotes the double loop space of the 3–connected cover of the 3–sphere. The methods are to translate the structure of H*(Bn;F[t±1]) to one concerning the structure of the homology of certain function spaces where the answer is computed.

Keywords

homotopy groups, braid groups, descending central series, loop spaces

Mathematical Subject Classification

Primary: 20F14, 20F36, 20F40, 52C35, 55Q99

Secondary: 12F99

References
Publication

Received: 15 January 2004
Revised: 5 June 2005
Published: 29 January 2007

Authors
F R Cohen
Department of Mathematics
University of Rochester
Rochester NY 14627
USA
J Pakianathan
Department of Mathematics
University of Rochester
Rochester NY 14627
USA