COHOMOLOGY OF BRAIDS, PRINCIPAL CONGRUENCE SUBGROUPS AND GEOMETRIC REPRESENTATIONS

The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2, ℤ) as well as their analogues in the 3-strand braid group with local coefficients in symmetric powers of the natural symplectic representation. The resulting answers (1) correspond to certain modular forms in characteristic zero and (2) the cohomology of certain spaces in homotopy theory in characteristic p. The torsion is given in terms of the structure of a ‘p-divided power algebra’. The work is an extension of the work in Callegaro et al. [The cohomology of the braid group B3 and of SL2(ℤ) with coefficients in a geometric representation, Quart. J. Math. 64 (2013), 847–889] as well as extensions of a classical computation of Shimura to integral coefficients. The results here contrast the local coefficients such as that in Looijenga [Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel–Jacobi map, J. Algebraic Geom. 5 (1996), 135–150] and Tillmann [The representation of the mapping class group of a surface on its fundamental group in stable homology, Quart. J. Math. 61 (2010), 373–380].