Title: The cohomology of the braid group $B_3$ and $SL_2(Z)$ with
coefficients in a geometric representation
Abstract:
I will discuss a joint work with Fred Cohen and Mario Salvetti. This
talk addresses the cohomology of the third braid group B_3 and the
group SL_2(Z) with coefficients in the symmetric powers of the natural
symplectic representation. Calculations of this type have been basic
in several areas of mathematics such as number theory as well as
algebraic geometry. The results can be regarded as giving the
cohomology of certain mapping class groups with non-trivial local
coefficients.
In general, let M_{g,n} be an orientable surface of genus g with n
connected components in its boundary. Isotopy classes of Dehn twists
around simple loops represents the braid group B_{2g+1} in the
symplectic group Aut(H1(M_{g,n}); <>) of all automorphisms preserving
the intersection form. In the case g = 1, n = 1 the symplectic group
equals SL_2(Z) and we extend the above representation to the symmetric
power M = Z[x; y]. Our result is the complete integral computation of
H^*(B_3;M) and H^*(SL2(Z);M).
These groups have a description in terms of a variation of the so
called "divided polynomial algebra". Unexpectedly, we found a strong
relation between the computed cohomology and some spaces which were
constructed in a completely different framework and which are basic to
the growth of p-torsion in the homotopy groups of spheres.