Mapping class groups and function spaces (Q2752867)

Let \(\Gamma^k\) be the braid group of the sphere on \(k\) strings, also referred to as the mapping class group of the sphere with \(k\) marked points. Its (co)homology with prime coefficients is determined in terms of the (co)homology of \(B\text{SO}_3\), the classifying space of the \(3\times 3\) special orthogonal group, and in terms of the (co)homology of the braid group (at the prime 2) respectively in terms of filtration quotients of certain configuration spaces related to the double loop space of the 2-sphere (at odd primes). Also, in the odd primary case, coefficients in the sign representation are considered. Some explicit calculations of Euler-Poincaré series are given at the prime \(3\) when \(n= 3,4,5,6\).NEWLINENEWLINENEWLINELet \(\Delta_g\) denote the hyperelliptic mapping class group, i.e. the stabilizer of the hyperelliptic involution in the mapping class group of a genus \(g\) surface. As \(\Gamma^{2g+2}\) is the quotient of \(\Delta_g\) by the hyperelliptic involution, the (co)homology of \(\Delta_g\) with any coefficients in which \(2\) is inverted can be deduced.NEWLINENEWLINENEWLINEThe main idea is to work with an explicit model \(E\) for the classifying space of \(\Gamma^k\). Consider the \(S^2\) fibration \(B\text{SO}_2\to B\text{SO}_3\). Then \(E\) is the space of fiberwise, unordered \(k\)-tuples in \(S^2\). Sections 3 to 5 explain and generalize this construction. Configuration space techniques are the main input. Sections 6 to 10 are devoted to calculations of the (co)homology, separating the even and odd primary case.NEWLINENEWLINENEWLINEThe present paper is up to minor changes identical with the authors' Mathematica Gottingensis preprint, Heft 5 (1989). Some weaker results have subsequently been proved by \textit{N. Kawazumi} [Topology Appl. 76, No. 3, 203-216 (1997; Zbl 0892.57008)] using more geometric methods.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00013].