德恩扭转的不动点弗洛尔上同调I：分裂公式
Fixed Point Floer Cohomology of Dehn Twists I: Splitting Formulas
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中文总结 AI 辅助
该论文为系列首篇，研究德恩扭转的不动点弗洛尔上同调。利用新限制结果，开发高维情形下计算工具，证明其乘积和微分在余链层面可分解为局部和莫尔斯理论贡献，还提及对未来工作的应用与探讨。
中文摘要 AI 辅助
本文是我们早期研究不动点弗洛尔上同调中裤对积的系列论文中的第一篇。在早期工作中，我们已完全计算出亏格大于或等于2的曲面上德恩扭转的该乘积，并用于计算节点曲线的一个版本的（小）“量子上同调”。在当前工作中，我们为所有更高维的德恩扭转情形开发计算不动点弗洛尔上同调及相关乘积的工具。对于围绕刘维尔域中拉格朗日球面的迭代德恩扭转，利用J - 全纯曲线的新限制结果，我们表明不动点弗洛尔上同调的乘积和微分在余链层面可分解为局部和莫尔斯理论贡献。局部贡献有望恢复\(S^n\)（扭曲）环空间同调的有限扇区及相关的查斯 - 沙利文乘积，我们将在未来工作中详细研究。我们还讨论了一些对未来工作的直接应用和有趣之处。
英文摘要
This paper is the first in a series following on our earlier work [arXiv:2205.14516, arXiv:2307.08180] studying the pair-of-pants product on fixed point Floer cohomology. In [arXiv:2205.14516, arXiv:2307.08180] we fully computed this product for Dehn twists on surfaces of genus greater or equal to 2, and used it to compute a version of the (small) "quantum cohomology" for nodal curves. In the present work, we develop tools for computing the fixed point Floer cohomology and the associated product in the case of Dehn twists in all higher dimensions: for iterated Dehn twists around a Lagrangian sphere in a Liouville domain, we show that the product and differential on the fixed point Floer cohomology split into local and Morse-theoretic contributions on the level of cochains, using some new confinement results for J-holomorphic curves. The local contributions are expected to recover a finite sector of the homology of (twisted) loop spaces of $S^n$ along with an associated Chas-Sullivan product, which we will examine in detail in future work. We also discuss some immediate applications and curiosities for future work.
