THE p-PERIODICITY OF THE MAPPING CLASS GROUP AND THE ESTIMATE OF ITS p-PERIOD

We determine completely the primes p for which the Farrell-Tate cohomology of the mapping class group Tg is /»-periodic. We also estimate the p-period of a p-periodic T„ . For T any group of finite virtual cohomological dimension (vcd) and a prime p, we say that the group Y has p-periodic cohomology if there exists a positive integer d such that the Farrell-Tate cohomology groups //'(r; M) and Hl+d(T; M) have naturally isomorphic p-primary components for all i e Z and ZT-modules M. The p-period of T is defined as the least value of d [Bl]. Recall that the mapping class group Yg is defined to be the group of path components of orientation-preserving homeomorphisms of the orientable closed surface Sg of genus g. We always assume g > 1 . It is well known that the mapping class group Tg is of finite vcd and the vcd(r?) = Ag 5 [H]. In this paper, we determine completely the primes p for which Tg is pperiodic. Furthermore, we estimate the p-period of a p-periodic Yg by using the p-period of a metacyclic subgroup of Yg as a lower bound and a homogeneous Chern class polynomial of the canonical homology representation Fg —> GL(2g, Z) as an upper bound. The main results are as follows: Theorem 1. (a) The mapping class group Tg is never 2-periodic. (b) The mapping class group Ykp+i is always p-periodic when i / 1 mod(p) where p is an odd prime and k > 0. (c) The mapping class group Ykp+l is p-periodic if and only if the interval [(2k + 3)1 p, (2k + 2)l(p 1)] does not contain an integer and k ± 0, -1 mod(p) where p is an odd prime. In particular, Ykp+i can be p-periodic only when k < (p2 -5)/2. Theorem 2. // k ^ 0 mod(p) and p > 2, then Y(p_-[-}(kp_k_2y2 is p-periodic and the p-period of Tfj,-\)(kp-k-2)ii & a multiple of 2(p 1). Moreover, if k < (p l)/2, the p-period of T(p_mp-k-2)i2 equals 2(p 1). Theorem 3. If 3 < d and d divides p 1, then r(;,_1)(¿_2)/2 is p-periodic and the p-period of Y^p_l^d_2y2 is a multiple of 2d. Received by the editors February 28, 1991. 1991 Mathematics Subject Classification. Primary 57R20, 20F38, 20J10. ©1992 American Mathematical Society 0002-9939/92 $1.00+ $.25 per page