Artin's Braid Group and the Homology of Certain Subgroups of the Mapping Class Group F. R. Cohen* Department of Mathematics University of Rochester Rochester, NY 14627 U.S.A. 1 Statement of Results By work of B. Wajnryb [21], the mapping class group k , k 0,1, is generated by 2g + 1 Dehn twists a set of generators is given by Dehn twists a i , . . . , ctg,/3i,..., j3g, and 6 about embedded circles with the same names as in the picture below. The group Ag is the subgroup of T^0 generated by c*i,... ,a 5 ,/?i,... ,Pg, and ag+i. In case g = 2, then A2 = T^Q (Birman and Hilden [4]). It is easy to see that Ag is strictly smaller than r ° 0 if g 3 (Birman and Hilden [4]). There is a non-split central extension i ^ z/2 A , -+ r^f0+2 - I by Birman and Hilden [4]. The group TQ 0 has elements of infinite order if n 4. Many of the results given below about Ag and T^o w e r e stated in Cohen [8] without proof. Theorem 1.1 The reduced integral homology of TQ0 is all p-torsion for primes p with p n. The reduced integral homology of Ag is all p-torsion with p 2g + 2. * Partially supported by a National Science Foundation grant 6
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