标题： 链级Koszul对偶性在Gravity操作符和超交换操作符之间 显示英文标题

标题： Chain level Koszul duality between the Gravity and Hypercommutative operads

摘要： 设$\overline{\mathcal{M}}_{0,n+1}$为带有$(n+1)$个标记点的零亏格稳定曲线的模空间。 该集合$\overline{\mathcal{M}}=\{\overline{\mathcal{M}}_{0,n+1}\}_{n\geq 2}$在复射影代数簇的范畴中构成一个操作子；其上同调$Hycom= H_*(\overline{\mathcal{M}})$称为超交换操作子。 在本文中我们构造了一个超交换操作子的链模型，即一个链复形的操作子$C_*^{dual}(\overline{\mathcal{M}})$，它与奇异链操作子$C_*(\overline{\mathcal{M}})$弱等价。 我们证明$C_*^{dual}(\overline{\mathcal{M}})$是巴构造$B(grav)$的线性对偶，其中$grav$是基于无基点的卡蒂的重力操作符的链模型。 这表明重力操作符和超交换操作符在链层面上也是 Koszul 对偶，改进了 Getzler 的一个先前结果。 该构造是拓扑的，因为$C_*^{dual}(\overline{\mathcal{M}})(n)$是与$\overline{\mathcal{M}}_{0,n+1}$的正则 CW 分解相关的细胞复形。 显示英文摘要

摘要： 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 $\overline{\mathcal{M}}_{0,n+1}$.