Periodicity of the punctured mapping class group

The main result is that the punctured mapping class group Td (i greater than or equal to 1, g greater than or equal to 1) has periodic cohomology; furthermore, the period is always 2. We present a proof which involves the Yagita invariant and the Chem class of the representation of a subgroup in Gamma (i)(g) (i greater than or equal to 1, g greater than or equal to 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 Gamma (0)(g) to the case Gamma (i)(g) (i greater than or equal to 1, g greater than or equal to 1). (C) 2001 Elsevier Science B.V. All rights reserved. MSC. Primary 55; 20.