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Fiber Floer cohomology and conormal stops
Symplectic Geometry · Submitted 5 Dec 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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 $x\in M_K$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $S$ be a closed orientable spin manifold. | background 1.00 | 0.05 | 0.01 | asserted 0.77 |
| s1 | Let $K \subset S$ be a submanifold and denote its complement by $M_K$. | background 1.00 | 0.11 | 0.01 | asserted 0.53 |
| s2 | 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. | result 0.84 | 0.96 | 0.95 | asserted 1.00 |
| s3 | As an application, we restrict to codimension 2 spheres $K \subset S^n$ where $n = 5$ or $n\geq 7$. | contribution 0.62 | 0.76 | 0.03 | asserted 0.76 |
| s4 | 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 $K$. | result 0.92 | 0.94 | 0.10 | asserted 1.00 |
| s5 | 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 $x\in M_K$. | result 0.53 | 0.89 | 0.12 | asserted 1.00 |
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Original 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 $x\in M_K$.
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2 arXiv papers cite this in 3 sentences
Papers citing it · 2 sentences without a confident label
Cites it as background 1 paper
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To the end, we mention also the work of J. Asplund [ref] which studied $ℒ^∗N$ using partially wrapped Floer homology in the spirit of wrapped Fukaya categories of Liouville sectors from [ref] .
A cord algebra for tori in three-space · Algebraic invariants from conormal lifts · 0.92
What it cites · 22 sentences without a confident label
Uses the method or tool of 3 papers
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Consider the Moore loop space of $M_K$, based at $ξ$ [equation] We use a cubical model for chains of based loops as in [ref] .
Duality between Lagrangian and Legendrian invariants · Based loops on $M_K$ · 0.89
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A specific perturbation scheme involving anchored curves is constructed in [ref] , and we fix such perturbation scheme so that all relevant moduli spaces are transversely cut out.
Holomorphic curves for Legendrian surgery · Wrapped Floer cohomology without Hamiltonian · 0.98
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We define $CW^∗__K(F,F)$ using a surgery approach similar to [ref] and [ref] (see <ref> for details).
A complete knot invariant from contact homology · Introduction · 0.84
Cites as background 4 papers
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Using notation as in [ref] , we are interested in the moduli spaces of $J$-holomorphic disks which are denoted by $ℳ^fi(c;κ)$, $ℳ^sy(c;κ)$ and $ℳ^pb(c;κ)$.
Duality between Lagrangian and Legendrian invariants · $A_∞$-structure and moduli space of disks · 0.85
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Consider the wrapped Fukaya category of $T^∗M_K$ [ref] .
A wrapped Fukaya category of knot complement · Relation to other results · 0.84
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Wrapped Floer cohomology without Hamiltonian has been studied in e.g. [ref] and in particular it is useful in proving various surgery formulas involving the wrapped Floer cohomology [ref] .
Holomorphic curves for Legendrian surgery · Wrapped Floer cohomology without Hamiltonian · 0.87
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