Fiber Floer cohomology and conormal stops

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 ${\Lambda }_K$ 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 $n\ge 7$. 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 $x\in M_K$.

55 pages, 19 figures. v3: Final version to appear in Journal of Symplectic Geometry

References in corpus (3)

• Lectures on Symplectic Field Theory
• Holomorphic curves for Legendrian surgery
• Equivariant Spectral Sequences for Local Coefficients

Cited by in corpus (3)

• Simplicial descent for Chekanov-Eliashberg dg-algebras
• Chekanov-Eliashberg dg-algebras for singular Legendrians
• Tangle contact homology