On the Yagita invariant of mapping class groups
type: Item
instance of: scholarly article
title: On the Yagita invariant of mapping class groups
zbMATH Open document ID: 0828.57011
DOI: 10.1016/0040-9383(94)90028-0
author: Guido Mislin
author: Yining Xia
author: Henry H. Glover
published in: Topology
publication date: 1994-09-05T00:00:00Z
review text: For a prime \(p\), the Yagita invariant \(p (\Gamma)\) of a group \(\Gamma\) of finite virtual cohomological dimension is a natural number defined by considering and comparing, in a certain natural way, the sizes of the images of the restriction maps of the cohomology of \(\Gamma\) to the cohomology, with coefficients mod \(p\), of the various subgroups of order \(p\) in \(\Gamma\). The interest in \(p (\Gamma)\) stems from the fact that it provides a lower bound for the dimension of a complex which admits a certain action of the group \(\Gamma\). Also, the invariant \(p (\Gamma)\) agrees with the \(p\)-period of a \(p\)-periodic group, i.e. a group with \(p\)-periodic Farrell cohomology. Now let \(\Gamma = \Gamma_g\) be the mapping class group (isotopy classes of diffeomorphisms) of a closed orientable surface of genus \(g\). For odd primes \(p\) and \(p\)-periodic \(\Gamma_g\) the authors completely determined the Yagita invariant or \(p\)-period \(p (\Gamma_g)\) in a previous paper. The present paper is devoted to the computation of \(p (\Gamma_g)\) is the remaining cases where \(p (\Gamma_g)\) is not \(p\)- periodic (which implies that \(g\) is congruent to 1 mod \(p)\). The obtained results are complete in the case of a regular (odd) prime, in the remaining cases of irregular primes some additional condition is needed. In the proofs, lower resp. upper bounds for \(p (\Gamma_g)\) are obtained by studying representations of subgroups of order \(p\) of \(\Gamma_g\) on the space of holomorphic differentials of a Riemann surface of genus \(g\), resp. the action of \(\Gamma_g\) on a suitable submanifold of the Teichmüller space of surfaces of genus \(g\).
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 20F34
zbMATH DE Number: 633637
zbMATH Keywords: group of finite virtual cohomological dimension
zbMATH Keywords: \(p\)-periodic group
zbMATH Keywords: Yagita invariant
zbMATH Keywords: mapping class group
zbMATH Keywords: closed orientable surface
zbMATH Keywords: irregular primes
zbMATH Keywords: space of holomorphic differentials
zbMATH Keywords: Teichmüller space
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: 0040-9383(94)90028-0
OpenAlex ID: W2010953510
description: scientific article; zbMATH DE number 633637