On operadic open-closed maps in characteristic $p$
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913349963350016 |
|---|---|
| author | Chen, Zihong |
| author_facet | Chen, Zihong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06183 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On operadic open-closed maps in characteristic $p$ Chen, Zihong Symplectic Geometry K-Theory and Homology 53D45, 53D37, 16E40 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. |
| title | On operadic open-closed maps in characteristic $p$ |
| topic | Symplectic Geometry K-Theory and Homology 53D45, 53D37, 16E40 |
| url | https://arxiv.org/abs/2402.06183 |