Chain level Koszul duality between the Gravity and Hypercommutative operads
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916507568570368 |
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| author | Rossi, Tommaso Salvatore, Paolo |
| author_facet | Rossi, Tommaso Salvatore, Paolo |
| contents | Let $\overline{\mathcal{M}}_{0,n+1}$ be the moduli space of genus zero stable curves with $(n+1)$-marked points. The collection $\overline{\mathcal{M}}=\{\overline{\mathcal{M}}_{0,n+1}\}_{n\geq 2}$ forms an operad in the category of complex projective varieties; its homology $Hycom= H_*(\overline{\mathcal{M}})$ is called the Hypercommutative operad. In this paper we construct a chain model for the hypercommutative operad, i.e. an operad of chain complexes $C_*^{dual}(\overline{\mathcal{M}})$ which is weakly equivalent to the operad of singular chains $C_*(\overline{\mathcal{M}})$. We prove that $C_*^{dual}(\overline{\mathcal{M}})$ is the linear dual of the bar construction $B(grav)$, where $grav$ is a chain model of the gravity operad based on cacti without basepoint. This shows that the Gravity and Hypercommutative operad are Koszul dual also at the chain level, refining a previous result of Getzler. The construction is topological, since $C_*^{dual}(\overline{\mathcal{M}})(n)$ is the cellular complex associated to a regular CW-decomposition of $\overline{\mathcal{M}}_{0,n+1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03474 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chain level Koszul duality between the Gravity and Hypercommutative operads Rossi, Tommaso Salvatore, Paolo Algebraic Topology Algebraic Geometry Let $\overline{\mathcal{M}}_{0,n+1}$ be the moduli space of genus zero stable curves with $(n+1)$-marked points. The collection $\overline{\mathcal{M}}=\{\overline{\mathcal{M}}_{0,n+1}\}_{n\geq 2}$ forms an operad in the category of complex projective varieties; its homology $Hycom= H_*(\overline{\mathcal{M}})$ is called the Hypercommutative operad. In this paper we construct a chain model for the hypercommutative operad, i.e. an operad of chain complexes $C_*^{dual}(\overline{\mathcal{M}})$ which is weakly equivalent to the operad of singular chains $C_*(\overline{\mathcal{M}})$. We prove that $C_*^{dual}(\overline{\mathcal{M}})$ is the linear dual of the bar construction $B(grav)$, where $grav$ is a chain model of the gravity operad based on cacti without basepoint. This shows that the Gravity and Hypercommutative operad are Koszul dual also at the chain level, refining a previous result of Getzler. The construction is topological, since $C_*^{dual}(\overline{\mathcal{M}})(n)$ is the cellular complex associated to a regular CW-decomposition of $\overline{\mathcal{M}}_{0,n+1}$. |
| title | Chain level Koszul duality between the Gravity and Hypercommutative operads |
| topic | Algebraic Topology Algebraic Geometry |
| url | https://arxiv.org/abs/2412.03474 |