MAPPING CLASS GROUPS 9 By inspection of the above presentations, there are natural quotient maps 7 : Bn —• TQ0, 7r : J9n S n and A : TQ0 S n . The kernel of n is the pure braid group Pn (Artin [2], Birman [3]). Magnus identifies the kernel of 7. It is isomorphic to Z X F n _i where Fn-\ is a free group on (n 1) generators. Write Kn for the kernel of A. Notice that there is a commutative diagram of short exact sequences (where nine copies of the trivial group have been deleted) Z x Fn-\ —• Pn —• Kn Z x Fn-i Bn -+ TQ0 I I* JA 1 * L n » L n . In what follows, our main computations will be to apply the Lyndon-Hochschild-Serre spectral sequence to the group extension 1 Kn —• I ^ Q —• E n —• 1. Thus we need to compute the cohomology of E n with coefficients in the cohomology of Kn this being the i?2-term of the Lyndon-Hochschild-Serre spectral sequence. 3 H*(K n Z) A computation of the integral cohomology of Kn as an algebra over the group ring of E n is given in this section. The method of proof is to show that a : Pn Kn induces a monomorphism in cohomology. Since the cohomology of Pn is understood and a* com- mutes with the En-action, Theorem 1.5 will follow. Thus we first recall the cohomology of Pn (Arnol'd [1] Cohen-Lada-May [11, pp.250-266]) in this last reference, the notation at-+ij is used for Aij in the following theorem. Theorem 3.1 The integral cohomology of Pn is torsion free with Poincare series (l + 0 ( l + 2t)---(l + ( n - l ) t ) and is generated as an algebra by one-dimensional classes Aij, n i j 1. A complete set of relations is given by 4 = 0 AijAik = Akj(Aik - A^) if j k. The action of E n is determined by crAlJ = A^^j) for a G S n with the convention that Aji Aij. A basis for Hq(Pn Z) is given by Aiin ... Aiqjq, 2i1i2-'iq n.
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