The Johnson homomorphism and the third rational cohomology group of the Torelli group
type: Item
instance of: scholarly article
title: The Johnson homomorphism and the third rational cohomology group of the Torelli group
zbMATH Open document ID: 1066.57021
author: Takuya Sakasai
published in: Topology and its Applications
publication date: 2005-03-08T00:00:00Z
review text: The Torelli group \(\mathcal{I}_g\) is defined by taking the mapping class group of a closed oriented surface of genus \(g\) at least \(3\); \(\mathcal{I}_g\) consists of those elements of the mapping class group which act trivially on the first homology group, \(H\), of the surface. \textit{D. Johnson} [Math. Ann. 249, 225--242 (1980; Zbl 0409.57009)] defined a homomorphism with domain \(\mathcal{I}_g\) and range a three-fold product of \(H\) modulo a certain subgroup isomorphic to \(H\). Following \textit{D. Johnson} [Topology 24, 127--144 (1985; Zbl 0571.57010)] and \textit{R. Hain} [J. Am. Math. Soc. 10, 597--651 (1997; Zbl 0915.57001)], the author uses the induced homomorphism on homology to study the third rational homology group of \(\mathcal{I}_g\).
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 2142282
zbMATH Keywords: Torelli group
zbMATH Keywords: mapping class group
zbMATH Keywords: rational homology
zbMATH Keywords: Johnson homomorphism
reviewed by: Martin D. Crossley
MaRDI profile type: Publication
full work available at URL: j.topol.2004.08.002
OpenAlex ID: W2139336505
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DOI: 10.1016/J.TOPOL.2004.08.002
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description: scientific article; zbMATH DE number 2142282