On the Torelli Lie algebra. (English) Zbl 1516.57028
Let \(\Sigma_{g,1}\) denote a compact oriented surface of genus \(g\) with one boundary component. The isotopy classes of orientation-preserving diffeomorphisms of \(\Sigma_{g,1}\) which act as the identity on \(H_1(\Sigma_{g,1}; {\mathbb{Z}})\) form a group \(T_{g,1}\) called the Torelli group of \(\Sigma_{g,1}\). Let \({\mathbf{t}}_{g,1}\) denote the Mal’cev Lie algebra associated to the group \(T_{g,1}\). R. Hain has completely determined \({\mathbf{t}}_{g,1}\) and proved that it is quadratic, if the genus is at least \(4\), see [Contemp. Math. 150, 75–105 (1993; Zbl 0831.57005); J. Am. Math. Soc. 10, No. 3, 597–651 (Zbl 0915.57001); J. Topol. 8, No. 1, 213–246 (2015; Zbl 1318.14028)]. In this paper, the authors investigate \({\mathbf{t}}_{g,1}\) further. They prove that stably, \({\mathbf{t}}_{g,1}\) is Koszul. They also consider the geometric Johnson homomorphism of \({\mathbf{t}}_{g,1}\), and show that the kernel of that homomorphism consists only of trivial \(\mathrm{Sp}_{2g}({\mathbb{Z}})\)-representations lying in the centre.
Reviewer: Marja Kankaanrinta (Helsinki)
MSC:
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 57S05 | Topological properties of groups of homeomorphisms or diffeomorphisms |
| 18M70 | Algebraic operads, cooperads, and Koszul duality |
| 20G05 | Representation theory for linear algebraic groups |
Keywords:
Torelli group; Torelli Lie algebra; Mal’cev completion; compact oriented surface; geometric Johnson homomorphism; Koszul dualityReferences:
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