On the Farrell cohomology of mapping class groups

Abstract

For Γ any group of finite virtual cohomological dimension and a prime p, we say that Γ is p-periodic, if there exists a positive integer k such that the Farrell cohomology groups Ĥ(Γ;M) and Ĥ(Γ;M) have naturally isomorphic p-primary components for all i ∈ Z and ZΓ-modules M . The p-period of Γ is defined as the least value of k (cf. [B]). For instance, if Γ is p-torsion free, then Γ is p-periodic of period one. The mapping class group, Γg, is defined to be the group of path components of the group of orientation preserving homeomorphisms of the oriented closed surface Sg of genus g. For instance, Γ1 ∼= SL(2,Z) and the cohomology is well known and easy to compute in this case. By writing SL(2,Z) as an amalgamated product of Z/4 and Z/6 over Z/2, one finds Ĥ(Γ1;Z) ∼= (Z/12)[x, x−1]