Torelli group action on the configuration space of a surface. (English) Zbl 1512.55016
Let \(X\) be a complex manifold and let \(F_n(X)\) denote the \(n\)-point configuration space of \(X.\)
From [B. Totaro, Topology 35, No. 4, 1057–1067 (1996; Zbl 0857.57025)], there is a spectral sequence which produces a weight filtration on \(H^\bullet(F_n(X)),\) whose graded pieces \(\operatorname{gr}^W_\bullet H^\bullet(F_n(X))\) only depend on \(H^\bullet(X).\) Moreover the mapping class group \(\operatorname{Mod}(X)\) acts on \(H^\bullet(F_n(X))\) preserving the weight filtration and on \(\operatorname{gr}^W_\bullet H^\bullet(F_n(X))\) through its representation on \(H^\bullet(X).\)
The author proves that what is true for \(\operatorname{gr}^W_\bullet H^\bullet(F_n(X))\) is not true, in general, for \(H^\bullet(F_n(X)).\)
This is done by showing that the Torelli group of a closed orientable surface \(\Sigma\) of genus \(\geq 3\) acts non-trivially on the \(\mathfrak{S}_3\)-invariant part \(H^3(F_3(\Sigma))^{\mathfrak{S}_3}\) of the cohomology of the \(3\)-point configuration space of \(\Sigma.\)
From [B. Totaro, Topology 35, No. 4, 1057–1067 (1996; Zbl 0857.57025)], there is a spectral sequence which produces a weight filtration on \(H^\bullet(F_n(X)),\) whose graded pieces \(\operatorname{gr}^W_\bullet H^\bullet(F_n(X))\) only depend on \(H^\bullet(X).\) Moreover the mapping class group \(\operatorname{Mod}(X)\) acts on \(H^\bullet(F_n(X))\) preserving the weight filtration and on \(\operatorname{gr}^W_\bullet H^\bullet(F_n(X))\) through its representation on \(H^\bullet(X).\)
The author proves that what is true for \(\operatorname{gr}^W_\bullet H^\bullet(F_n(X))\) is not true, in general, for \(H^\bullet(F_n(X)).\)
This is done by showing that the Torelli group of a closed orientable surface \(\Sigma\) of genus \(\geq 3\) acts non-trivially on the \(\mathfrak{S}_3\)-invariant part \(H^3(F_3(\Sigma))^{\mathfrak{S}_3}\) of the cohomology of the \(3\)-point configuration space of \(\Sigma.\)
Reviewer: Angelina Zheng (Roma)
MSC:
| 55R80 | Discriminantal varieties and configuration spaces in algebraic topology |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
Citations:
Zbl 0857.57025References:
| [1] | Bianchi, A., Splitting of the homology of the punctured mapping class group, J. Topol.13 (2020) 1230-1260. · Zbl 1530.55004 |
| [2] | A. G. Gorinov, A purity theorem for configuration spaces of smooth compact algebraic varieties, preprint (2017), arXiv:1702.08428. |
| [3] | Johnson, D., An abelian quotient of the mapping class group \(\mathcal{I}_g\), Math. Ann.249 (1980) 225-242. · Zbl 0409.57009 |
| [4] | Looijenga, E., Cohomology of \(\mathcal{M}_3\) and \(\mathcal{M}_3^1\), in Mapping Class Groups and Moduli Spaces of Riemann Surfaces (Göttingen, 1991/Seattle, WA, 1991), , Vol. 150 (Amer. Math. Soc., 1993), pp. 205-228. · Zbl 0814.14029 |
| [5] | Moriyama, T., The mapping class group action on the homology of the configuration spaces of surfaces, J. Lond. Math. Soc. (2)76 (2007) 451-466. · Zbl 1158.55020 |
| [6] | Totaro, B., Configuration spaces of algebraic varieties, Topology35 (1996) 1057-1067. · Zbl 0857.57025 |
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