Periodicity of the punctured mapping class group
type: Item
instance of: scholarly article
title: Periodicity of the punctured mapping class group
zbMATH Open document ID: 0964.57017
DOI: 10.1016/S0022-4049(99)00112-7
author: Qin Lu
published in: Journal of Pure and Applied Algebra
publication date: 2001-07-13T00:00:00Z
review text: Let \(\Gamma_g^i\) be the punctured mapping class group with \((i\geq 1, g\geq 1)\), and \(\Gamma_g^0\) be the unpunctured mapping class group, where \(g\) is the genus of the closed oriented two-manifold \(S_g\). The work of \textit{H. H. Glover, G. Mislin} and \textit{Y. Xia} [Topology 33, No. 3, 557-574 (1994; Zbl 0828.57011)] shows that \(\Gamma_g^0\) does not have periodic cohomology in general and \(\Gamma_g^0\) is never 2-periodic in group cohomology for \(g > 0\). The aim of paper under review is to prove that \(\Gamma_g^i\) has periodic cohomology with period 2. The result indicates one of the differences between the punctured mapping class group and the unpunctured mapping class group. The method used to prove the main result on periodicity is related to the Yagita invariant which is a generalization of the \(p\)-period of \(\Gamma_g^i\) if \(\Gamma_g^i\) has \(p\)-torsion. The period of the cohomology of a group is the least common multiple of all the \(p\)-periods where \(p\) ranges over the primes such that the group has \(p\)-torsion. Then the author shows that the Yagita invariant (as the \(p\)-period of \(\Gamma_g^i\)) is equal to 2 if \(\Gamma_g^i\) has \(p\)-torsion. The proof follows by induction on \(i\). (1) Suppose that \(\Gamma_g^1\) has \(p\)-torsion. Hence \(\Gamma_g^1\) has \(p\)-periodic cohomology (Theorem 1.2 of the paper) by a short exact sequence relating \(\Gamma_g^0\) and \(\pi_1(S_g)\) and by the relation on (virtual) cohomology dimensions; furthermore, the Yagita invariant \(Y(\Gamma_g^1, p) = 2\) provides the period 2 property by Theorem 1.4 of the paper. The calculation of \(Y(\Gamma_g^1, p)\) is based on the (weakly) homotopy equivalence between the classifying space of \(\Gamma_g^1\) and classifying space of Diff\(^+(S_g, *)\) for \(g > 1\); and the lifting of an arbitrary subgroup of \(\Gamma_g^1\) of order \(p\) to get representations into \(GL_2^+(R)\) and factorization to \(SO(2)\) (\(BGL_2^+(R) = BSO(2)\)) gives the relati…
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 1563014
zbMATH Keywords: periodicity of cohomology group
zbMATH Keywords: Yagita invariant
zbMATH Keywords: virtual cohomology dimension
MaRDI profile type: Publication
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Wikidata QID: Q126388589
full work available at URL: s0022-4049(99)00112-7
OpenAlex ID: W2071807170
description: scientific article; zbMATH DE number 1563014