On the non-vanishing of the powers of the Euler class for mapping class groups
type: Item
instance of: scholarly article
title: On the non-vanishing of the powers of the Euler class for mapping class groups
zbMATH Open document ID: 1490.57015
author: Rita Jiménez Rolland
author: Solomon Jekel
published in: Arnold Mathematical Journal
publication date: 2021-12-16T00:00:00Z
review text: Let \(\Gamma_{g}^{1}\) be the mapping class group of a surface \(\Sigma_{g}^{1}\) of genus \(g\geq 1\) and one marked point. Using the monomorphism \(\rho:\Gamma_{g}^{1}\hookrightarrow \mathrm{Homeo}_{+}(\mathbb{S}^{1})\) one pulls back the discrete universal Euler class \(\mathbf{E}\in H^{2} (\mathrm{Homeo}_{+}(\mathbb{S}^{1});\mathbb{Z})\) to \(E\in H^{2}(\Gamma_{g}^{1};\mathbb{Z})\). By naturality \(\mathbf{E}^{n}\) pulls back to \(E^{n}\in H^{2n}(\Gamma_{g}^{1};\mathbb{Z})\). The main theorem asserts that the classes \(E^{n}\) are nonzero for \(g\geq1\) and \(n\geq 1\). Furthermore, for \(n\geq g\) the subgroup of \(H^{2n}(\Gamma_{g}^{1};\mathbb{Z})\) generated by the class \(E^{n}\) is a finite cyclic group of order a multiple of \(4g(2g+1)\). The authors detect torsion classes by pulling back the Euler class to finite subgroups of \(\mathrm{Homeo}_{+}(\mathbb{S}^{1})\). The authors also prove a vanishing result for powers of the Euler class in the cohomology of pure mapping class groups. Namely, let \(1\leq i\leq k\), then the powers \(E^{n}_{i}\in H^{2n}(\Gamma_{g}^{k};\mathbb{Q})\) vanish for all \(n\geq g\).
Mathematics Subject Classification ID: 57K20
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 7446993
zbMATH Keywords: mapping class group
zbMATH Keywords: cohomology
zbMATH Keywords: Euler class
zbMATH Keywords: torsion
reviewed by: Daniel Juan Pineda
MaRDI profile type: Publication
full work available at URL: s40598-020-00159-3
OpenAlex ID: W3084261994
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DOI: 10.1007/S40598-020-00159-3
description: scientific article; zbMATH DE number 7446993