Periodicity of the punctured mapping class group

The main result is that the punctured mapping class group Γgi (i≥1, g≥1) has periodic cohomology; furthermore, the period is always 2. We present a proof which involves the Yagita invariant and the Chern class of the representation of a subgroup in Γgi (i≥1, g≥1). Using the main result, we can calculate the p-torsion of the Farrell cohomology for some special values of g and i. To do this, we extend the definition of the fixed point data as well as the conjugation theorem known for the case Γg0 to the case Γgi (i≥1, g≥1).