The p-torsion of the Farrell—Tate cohomology of the mapping class group Γp − 1

The mapping class group of a closed orientable surface of genus g, denoted Γg, is defined as the group of path components of the group of orientation preserving diffeomorphisms of the surface. We completely calculate the p-torsion of the Farrell-Tate cohomology of Γp − 1. The Farrell-Tate and ordinary cohomologies of Γg coincide above the virtual cohomological dimension 4g − 5.The basic method is to describe for Zp⊂Γp − 1 the quotient group N(Z / p) / Z / p as a finite extension of the pure mapping class group, where N(·) is the normalizer in Γp−1. Then putting a result of Cohen about the cohomology of the pure mapping class group into the Lyndon- Hochschild-Serre spectral sequence, we obtain the ordinary (and Farrell-Tate) cohomology of the normalizer group. The result about the p-torsion of the Farrell-Tate cohomology of Γp−1 then follows.