On the cohomology of the hyperelliptic mapping class group

We show that for every odd prime p, the hyperelliptic mapping class group Δg has p-periodic cohomology and we determine the p-period. We also compute the Yagita invariant at the prime 2 for even genus, with partial results for odd genus. Finally, we compute the p-part of the Farrell cohomology of Δg for g=(p−1)/2 and g=p−1, which are the first two cases of Δg containing p-torsion. Our methods include an analysis of fixed point data and the development of a braid calculus for Δg.