Fiber Floer cohomology and conormal stops

TL;DR

This paper establishes a deep connection between wrapped Floer cohomology and Morse theory on loop spaces, with applications to knot invariants and Legendrian isotopy distinctions.

Contribution

It proves an isomorphism linking wrapped Floer cochains with Morse chains on loop spaces and applies this to relate Floer cohomology to knot invariants, revealing new distinctions in Legendrian knot theory.

Findings

Isomorphism between Floer cochains and Morse chains of loop spaces

Relation between Floer cohomology and Alexander invariants of knots

Non-isotopy of certain Legendrian links involving conormal spheres

Abstract

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 $\varLambda_K$ 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 $\varLambda_K \cup \varLambda_x$ is not Legendrian isotopic to $\varLambda_{\mathrm{unknot}} \cup \varLambda_x$ where $x\in M_K$.