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| Main Authors: | , |
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| Format: | Preprint |
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2019
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| Online Access: | https://arxiv.org/abs/1905.04499 |
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| _version_ | 1866916452054859776 |
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| author | Khoroshkin, Anton Willwacher, Thomas |
| author_facet | Khoroshkin, Anton Willwacher, Thomas |
| contents | The real locus of the moduli space of stable genus-zero curves with marked points, $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$, is known to be a smooth manifold and is the Eilenberg-MacLane spaces for the so-called pure Cactus groups. We describe the operad formed by these spaces in terms of a homotopy quotient of an operad of associative algebras. Using this model, we identify various Hopf models for the algebraic operad of chains and homologies of $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$. In particular, we show that the operad $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal. As an application of these operadic constructions, we prove that for each $n$, the cohomology ring $H^{\bullet}(\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}), {\mathbb{Q}})$ is a Koszul algebra, and that the manifold $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal for $n\geq 6$ but is a rational $K(π,1)$-space. Additionally, we describe the Lie algebras associated with the lower central series filtration of the pure Cactus groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1905_04499 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Real moduli space of stable rational curves revised Khoroshkin, Anton Willwacher, Thomas Algebraic Topology Quantum Algebra 55P48, 18D50, 20F36, The real locus of the moduli space of stable genus-zero curves with marked points, $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$, is known to be a smooth manifold and is the Eilenberg-MacLane spaces for the so-called pure Cactus groups. We describe the operad formed by these spaces in terms of a homotopy quotient of an operad of associative algebras. Using this model, we identify various Hopf models for the algebraic operad of chains and homologies of $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$. In particular, we show that the operad $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal. As an application of these operadic constructions, we prove that for each $n$, the cohomology ring $H^{\bullet}(\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}), {\mathbb{Q}})$ is a Koszul algebra, and that the manifold $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal for $n\geq 6$ but is a rational $K(π,1)$-space. Additionally, we describe the Lie algebras associated with the lower central series filtration of the pure Cactus groups. |
| title | Real moduli space of stable rational curves revised |
| topic | Algebraic Topology Quantum Algebra 55P48, 18D50, 20F36, |
| url | https://arxiv.org/abs/1905.04499 |