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The purpose of this article is to describe the integral cohomology of the braid group B3 and SL2 (Z)with local coefficients in a classical geometric representation given by symmetric powers of thenatural symplectic representation. These groups have a description in terms of the so-called ‘dividedpolynomial algebra’. The results show a strong relation between the torsion part of the computedcohomology and fibrations related to loop spaces of spheres

THE COHOMOLOGY OF THE BRAID GROUP B3 AND OF SL2(Z) WITH COEFFICIENTS IN A GEOMETRIC REPRESENTATION