Fixed Point Floer Cohomology of Dehn Twists I: Splitting Formulas

Abstract

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.