Introduction D. J. Benson F. R. Cohen Let M^k denote an orient able surface (i.e., differential manifold of dimension two) of genus g with k boundary components and n punctures. The mapping class group T^k is defined to be the discrete group TToDiff4"(M£k) where DifT4" denotes the group of orientation preserving self-diffeomorphisms of the surface. In other words, an element of T™ fc can be thought of as an isotopy class of self-diffeomorphisms. Note that elements of T^k are allowed to permute the punctures, so that there is a surjective map Tn k » S n where £ n denotes the symmetric group on n letters. In fact, we are only interested in the case k = 0, but it should be noted that for k 0 the self-diffeomorphisms are required to fix the boundary pointwise. In this series of papers we study homological properties of the groups TQQ, 0 and a certain subgroup Ag of j 0 which was defined by Birman and Hilden [2] as follows. We express 0 as a double cover of the sphere S2, branched at 2g -f 2 points. Each self-diffeomorphism of S2 preserving the set of branch points lifts in two ways to a self- diffeomorphism of 0 , and Ag is defined to be the subgroup of T®0 represented by these lifts. Thus there is a (non-split) central extension l - Z / 2 - A , - r ^ 0 + 2 - l n L g,o The group Ag is generated by Dehn twists au ... ,a 5 ,/?i,... ,(3g and ag+i determined by circles as shown in the picture below. By results of Waynryb [16], « i , . . . ,%,/?i,.. . ,/3g and 8 give generators for T°j0. Thus A2 = T^o and Ag is a proper subgroup of Q for g 3. Dehn twists satisfy braid relations as follows. If the circles corresponding to a and r do not intersect then or ra, while if r and r intersect once transversally then arc TUT. 1
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