Torsion in the mapping class group and its cohomology
type: Item
instance of: scholarly article
title: Torsion in the mapping class group and its cohomology
zbMATH Open document ID: 0617.57005
DOI: 10.1016/0022-4049(87)90023-5
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let \(M_ g\) be the mapping class group of a closed orientable surface of genus g. It is known that \(M_ g\) has a torsionfree subgroup of finite index and finite cohomological dimension; especially, in large dimensions with respect to g, the cohomology \(H^*(M_ g)\) has p-torsion iff p is a torsion prime for \(M_ g\). On the other hand it has been proved by \textit{J. L. Harer} [Ann. Math., II. Ser. 121, 215-249 (1985; Zbl 0579.57005)] that \(H^ k(M_ g; {\mathbb{Z}})\) is independent of g for \(g>3k\), so there exist ''stable cohomology groups'' \(H^ k(M):=H^ k(M_ g; {\mathbb{Z}})\), for \(g\gg k\). Using Chern classes of the canonical representation \(M_ g\to Gl_{2g}({\mathbb{Z}})\) induced by the action on the homology of the surface the following main theorem is proved: The stable cohomology group \(H^{4k}(M)\) contains an element of order the denominator of \(B_{2k}/2k\), where \(B_{2k}\) is the 2kth Bernoulli number. These orders grow very rapidly and involve eventually any prime power infinitely often. ''In order to prove the main theorem we need results concerning the Chern classes of representations of cyclic groups and also about the existence of torsion in \(M_ g\). We get the latter by studying suitable \({\mathbb{Z}}/n {\mathbb{Z}}\)-actions on the surface and observing that these actions eventually stabilize in the sense that they always exist for \(g\gg n.''\) See also the following review.
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 20J99
zbMATH DE Number: 4000997
zbMATH Keywords: stable cohomology group of mapping class group of a surface
zbMATH Keywords: mapping class group of a closed orientable surface of genus g
zbMATH Keywords: Chern classes
zbMATH Keywords: Bernoulli number
zbMATH Keywords: Chern classes of representations of cyclic groups
author: Henry H. Glover
author: Guido Mislin
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90023-5
OpenAlex ID: W2067441117
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description: scientific article; zbMATH DE number 4000997