The Jones representation of genus 2 at the 4th root of unity and the Torelli group
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.
scientific article; zbMATH DE number 1879087
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