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The purpose of this article is to describe the integral cohomology of the braid group  B  3  and SL 2 (ℤ) with local coefficients in a classical geometric representation given by symmetric powers of the natural symplectic representation. These groups have a description in terms of the so-called ‘divided polynomial algebra’. The results show a strong relation between the torsion part of the computed cohomology and fibrations related to loop spaces of spheres.

THE COHOMOLOGY OF THE BRAID GROUP B 3 AND OF SL 2 (ℤ) WITH COEFFICIENTS IN A GEOMETRIC REPRESENTATION