Fiber Floer cohomology and conormal stops

Let $S$ be a closed orientable spin manifold. Let $K \subset S$ be a submanifold and denote its complement by $M_K$. In this paper we prove that there exists an isomorphism between partially wrapped Floer cochains of a cotangent fiber stopped by the unit conormal $\Lambda_K$ and chains of a Morse theoretic model of the based loop space of $M_K$, which intertwines the $A_\infty$-structure with the Pontryagin product. As an application, we restrict to codimension $2$ spheres $K \subset S^n$ where $n = 5$ or $n \geq 7$. Then we show that there is a family of knots $K$ so that the partially wrapped Floer cohomology of a cotangent fiber is related to the Alexander invariant of $K$. A consequence of this relation is that the link $\Lambda_K \cup \Lambda_x$ is not Legendrian isotopic to $\Lambda_\mathrm{unknot} \cup \Lambda_x$ where $x \in M_K$.

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Received 27 February 2020

Accepted 9 November 2020

Published 8 December 2021