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An Ozsváth--Szabó-type spectral sequence for links in $S^1\times S^2$

Jesse Cohen

Abstract

We show that there is a spectral sequence with $E^2$-page given by the Khovanov homology of a link in $S^1\times S^2$, as defined by Rozansky in arXiv:1011.1958, which converges to the Hochschild homology of an $A_\infty$-bimodule defined in terms of bordered Floer invariants. We also show that the homology algebras $H_*\mathfrak{h}_n$ of the algebras $\mathfrak{h}_n$ over which these bimodules are defined give nontrivial $A_\infty$-deformations of Khovanov's arc algebras $H_n$ for $n>1$.

An Ozsváth--Szabó-type spectral sequence for links in $S^1\times S^2$

Abstract

We show that there is a spectral sequence with -page given by the Khovanov homology of a link in , as defined by Rozansky in arXiv:1011.1958, which converges to the Hochschild homology of an -bimodule defined in terms of bordered Floer invariants. We also show that the homology algebras of the algebras over which these bimodules are defined give nontrivial -deformations of Khovanov's arc algebras for .
Paper Structure (27 sections, 35 theorems, 172 equations, 20 figures)

This paper contains 27 sections, 35 theorems, 172 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Organization and Results
  3. Background
  4. $A_\infty$-algebras and modules
  5. Hochschild homology
  6. Filtrations on $\mathcal{CH}(\mathcal{M})$ and their associated spectral sequences
  7. Type-$D$ structures
  8. Pairing $A_\infty$-modules and type-$D$ structures
  9. Filtered type-$\mathit{DA}$ bimodules
  10. Khovanov's arc algebras and bimodules
  11. Endomorphism Algebras of Type-$D$ structures
  12. Branched Arc Algebras
  13. Branched double covers
  14. The algebras
  15. Branched Bimodules and the Spectral Sequence
  16. ...and 12 more sections

Key Result

Lemma 1.1

Let $\mathcal{A}$ be an $A_\infty$-algebra over a commutative ring $\Bbbk$ and suppose that $\mathcal{M}$ is an $A_\infty$-bimodule over $\mathcal{A}$ equipped with a finite filtration, then there is a spectral sequence with $E^1$-page given by the Hochschild complex $\mathit{CH}(H_*\mathcal{A}[1],H

Figures (20)

  • Figure 1: The two crossingless matchings on 4 points are uniquely determined by the above diagrams.
  • Figure 2: A crossingless matching $b$ and the corresponding minimal saddle cobordism.
  • Figure 3: A diagram for a tangle $T\subset S^2\times[0,1]$ (left), its plat closure $p(T)$ by equatorial arcs (middle), and the cornered Seifert surface obtained from applying Seifert's algorithm to $p(T)$ (right). Here, the vertical lines in the left- and right-hand figures represent the projections of the equators of $S^2\times[0,1]$.
  • Figure 4: The genus 2 linear pointed matched circle.
  • Figure 5: Construction of a bordered Heegaard diagram for the 6-ended plat closure. Here, steps 1 through 4 are illustrated from left to right.
  • ...and 15 more figures

Theorems & Definitions (83)

  • Lemma 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark
  • Definition 2.2
  • Remark
  • Lemma 2.1: mescher2016primer
  • Corollary 2.2
  • proof
  • ...and 73 more