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*(Ω2S3⟨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
