The Jones representation of genus 2 at the 4th root of unity and the Torelli group
type: Item
instance of: scholarly article
title: The Jones representation of genus 2 at the 4th root of unity and the Torelli group
zbMATH Open document ID: 1019.57008
DOI: 10.1016/S0166-8641(01)00242-5
author: Yasushi Kasahara
published in: Topology and its Applications
publication date: 2003-03-10T00:00:00Z
review text: The Jones representation of the mapping class group of a closed surface of genus \(2\) is given by \textit{V. F. R. Jones} in [Ann. of Math. 126, 335-388 (1987; Zbl 0631.57005)]. The target of this representation is the group \(GL(5,{\mathbb{Z}}[t,t^{-1}])\). The paper under review describes the image of the Torelli group under the Jones representation when \(t=i\), the fourth root of unity. It turns out that the image of the Torelli group is isomorphic to the semidirect product of \({\mathbb{Z}}^8\) with \({\mathbb{Z}}_2\), where the action of \({\mathbb{Z}}_2\) on \({\mathbb{Z}}^8\) is given by the multiplication with \(-I\). Then the author discusses its relation with a certain Johnson homomorphism and the Rochlin invariant of a homology \(3\)-sphere.
reviewed by: Mustafa Korkmaz
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 57M60
zbMATH DE Number: 1879087
zbMATH Keywords: Jones representation
zbMATH Keywords: mapping class group
zbMATH Keywords: Torelli group
zbMATH Keywords: Johnson homomorphism
zbMATH Keywords: Rochlin invariant
MaRDI profile type: Publication
cites work: Q3812935
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full work available at URL: s0166-8641(01)00242-5
OpenAlex ID: W1986673816
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description: scientific article; zbMATH DE number 1879087