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Homology of hyperelliptic mapping class groups for surfaces. (English) Zbl 0892.57008

A closed surface \(F_g\) of genus \(g\) may be considered as a branch cover of the 2-sphere with \(2g+2\) branch points. The hyperelliptic involution \(j\) is an involution of \(F_g\) with \(2g+2\) fixed point. The hyperelliptic mapping class group \(\Delta_g\) is a subgroup of the mapping class group defined as the centralizer of \(j\). Its cohomology has previously been determined in [C.-F. Bödigheimer, F. R. Cohen and M. D. Peim, Mapping class groups and function spaces, preprint in Mathematica Gottingensis, Heft 5 (1989), unpublished]. The author’s approach in this paper is more elementary and geometric. His methods however also only recover the cohomology at primes \(p\) for \(p>g\).
Bödigheimer, Cohen and Peim start from the non-split central extension \(1\to\mathbb{Z}/2 \mathbb{Z}\to \Delta_g\to \Gamma^{2g+2} \to 1\) where \(\Gamma^{2g+2}\) denotes the braid group on \(2g+2\) strand of the sphere, that is the mapping class group of the sphere with \(2g+2\) marked points. Its cohomology is therefore that of \(\Delta_g\) for any trivial coefficients \(R\) containing \({1\over 2}\). They then proceed to describe the classifying space of \(\Gamma^k\) as the fiberwise configuration space of \(k\) unordered distinct points of the fibration \(S^2\to BSO(2) @>\mu>> BSO(3)\) where \(\mu\) is induced by the inclusion \(SO(2)\to SO(3)\). The analysis that then follows uses previous results of Bödigheimer and Cohen on stable and unstable splittings of mapping spaces, as well as further homotopy theoretic considerations when calculating the cohomology of \(\Gamma^k\) at different primes.
In the paper under review, in order to avoid problems arising in the presence of finite subgroups, the author considers surfaces with a boundary component and the associated hyperelliptic mapping class group \(\Delta_{g,1}\). The cohomology of \(\Delta_{g,1}\) (and hence of \(\Delta_g)\) is computed at primes \(p\) with \(p>g\) when the cohomology of Artin’s braid groups of the plane \(\beta_{2g+1}\) and \(\beta_{2g+2}\) are simple.
The key geometric result is the following. Let \(H_{g,1}\) be the moduli space of hyperelliptic curves of genus \(g\) with a marked point and a fixed tangent direction at the marked point. Then \(H_{g,1}\) is a classifying space for \(\Delta_{g,1}\). Let \(W_g\) be a subset of \(H_{g,1}\) where the marked point is a Weierstrass point. Then there are holomorphic isomorphisms \(W_g \simeq X_{2g+1}\) and \(H_{g,1}- W_g \simeq X_{2g+2}\) where \(X_n\) denotes the space of barycentered unordered configurations in \(\mathbb{C}^n\). As \(X_n\) is a classifying space for the braid group \(\beta_n\), the cohomology of \(\Delta_{g,1}\) can be deduced from that of the braid groups via the Gysin long exact sequence relating \(H_{g,1}\), \(X_{2g+1}\) and \(X_{2g+2}\).

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
57R50 Differential topological aspects of diffeomorphisms
20F36 Braid groups; Artin groups
20F38 Other groups related to topology or analysis
20J05 Homological methods in group theory
14H15 Families, moduli of curves (analytic)
14H52 Elliptic curves
14H55 Riemann surfaces; Weierstrass points; gap sequences

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References:

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