Homology of hyperelliptic mapping class groups for surfaces
type: Item
instance of: scholarly article
title: Homology of hyperelliptic mapping class groups for surfaces
zbMATH Open document ID: 0892.57008
DOI: 10.1016/S0166-8641(96)00105-8
author: Nariya Kawazumi
published in: Topology and its Applications
publication date: 1998-08-03T00:00:00Z
review text: 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 [\textit{C.-F. Bödigheimer}, \textit{F. R. Cohen} and \textit{M. D. Peim}, Mapping class groups and function spaces, preprint in Mathematica Gottingensis, Heft 5 (1989), unpublished].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 bra…
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57S05
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 14H15
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H55
zbMATH DE Number: 1022178
zbMATH Keywords: moduli space
zbMATH Keywords: configuration space
zbMATH Keywords: Weierstrass points
reviewed by: U. Tillmann
describes a project that uses: Mathematica
MaRDI profile type: Publication
cites work: Q5631231
cites work: Mapping class groups of low genus and their cohomology
cites work: Q5621787
cites work: Cohomology of braid spaces
cites work: Q3802260
cites work: On the Mapping Class Groups for Punctured Spheres, the Hyperelliptic Mapping Class Groups, SO(3), and Spin c (3)
cites work: Cohomologies of the group COS mod 2
cites work: Q5749984
cites work: The Nielsen realization problem
cites work: Characteristic classes of surface bundles
cites work: Q3216763
cites work: Cohomologies of braid groups
full work available at URL: s0166-8641(96)00105-8
OpenAlex ID: W2044838330
Wikidata QID: Q127011698
description: scientific article; zbMATH DE number 1022178