The cohomology of the braid group B_3 and of SL_2(Z) with coefficients in a geometric representation

TL;DR

This paper computes the integral cohomology of the braid group B_3 and SL_2(Z) with coefficients in geometric representations, revealing connections to divided polynomial algebra and loop space fibrations.

Contribution

It provides a detailed description of the cohomology with local coefficients for B_3 and SL_2(Z), highlighting the role of divided polynomial algebra and topological fibrations.

Findings

Cohomology groups expressed via divided polynomial algebra

Identification of torsion related to loop space fibrations

Explicit calculations of cohomology with geometric coefficients

Abstract

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 torsion part of the computed cohomology and fibrations related to loop spaces of spheres.