关于特征$p$中的算子开闭映射
On operadic open-closed maps in characteristic $p$
摘要：
考虑闭单调辛流形$(M,ω)$ . \cite{Gan2}构造了一个循环开闭映射，它从$M$的类循环同调到$M$的$S^1$ -等变量子上同调.本文证明了当$p$系数为mod时，Ganatra循环开闭映射在Hochschild同调的自然$\mathbb{Z}/p$ -Gysin型比较映射下与某个$\mathbb{Z}/p$ -等变开闭映射相容.在证明的基础上，给出了辛拓扑中开闭映射的同伦理论框架。这些将被用于在即将到来的工作\cite{Che}，以研究国防部$p$等变枚举不变量，如量子Steenrod操作。本文的主要内容是：1）（$\mathcal{A}_{\infty}$ -）循环对象的$\mathbb{Z}/p$-Gysin比较结果; 2）利用\cite{AGV}的操作Floer理论构造开闭映射，并对它的“$S^1$ -等变”性质给出新的解释;
英文摘要：
Consider a closed monotone symplectic manifold $(M,ω)$. \cite{Gan2} constructed a cyclic open-closed map, which goes from the cyclic homology of the Fukaya category of $M$ to the $S^1$-equivariant quantum cohomology of $M$. In this paper, we show that with mod $p$ coefficients, Ganatra's cyclic open-closed map is compatible with a certain $\mathbb{Z}/p$-equivariant open-closed map under the natural $\mathbb{Z}/p$-Gysin type comparison map for Hochschild homology. Along with the proof, this paper gives a new homotopy theoretic framework for studying open-closed maps in symplectic topology. These will be used in an upcoming work \cite{Che} to study mod $p$ equivariant enumerative invariants such as the Quantum Steenrod operations. The main insights of this paper are: 1) a $\mathbb{Z}/p$-Gysin comparison result for ($\mathcal{A}_{\infty}$-) cyclic objects, 2) a new construction of the open-closed map using operadic Floer theory of \cite{AGV}, which gives rise to a new interpretation of its `$S^1$-equivariant' property, and 3) comparison of the new construction with its classical counterpart.
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