The stable braid group and the determinant of the Burau representation

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 n-th braid group B_n with local coefficients obtained from the determinant of the Burau representation, H^*(B_n;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_infty; F[t^{+/-1}])-->H_*(Omega^2S^3<3>; F) for any field F where Omega^2S^3<3> denotes the double loop space of the 3-connected cover of the 3-sphere. The methods are to translate the structure of H_*(B_n;F[t^{+/-1}]) to one concerning the structure of the homology of certain function spaces where the answer is computed.