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A bordered HF- algebra for the torus

Robert Lipshitz, Peter Ozsváth, Dylan Thurston

Abstract

We describe a weighted $A_\infty$-algebra associated to the torus. We give a combinatorial construction of this algebra, and an abstract characterization. The abstract characterization also gives a relationship between our algebra and the wrapped Fukaya category of the torus. These algebras underpin the (unspecialized) bordered Heegaard Floer homology for three-manifolds with torus boundary, which will be constructed in forthcoming work.

A bordered HF- algebra for the torus

Abstract

We describe a weighted -algebra associated to the torus. We give a combinatorial construction of this algebra, and an abstract characterization. The abstract characterization also gives a relationship between our algebra and the wrapped Fukaya category of the torus. These algebras underpin the (unspecialized) bordered Heegaard Floer homology for three-manifolds with torus boundary, which will be constructed in forthcoming work.

Paper Structure

This paper contains 35 sections, 65 theorems, 324 equations, 20 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Weighted A-infinity algebras and modules
  3. Definition and first properties
  4. Gradings
  5. Definitions of the algebras via planar graphs
  6. The underlying associative algebra
  7. The definition of weighted A-infinity algebra
  8. Verifying the A-infinity relations
  9. First properties of the weighted A-infinity algebra of the torus
  10. From tiling patterns to immersions
  11. Gradings
  12. The big grading group
  13. The intermediate grading group
  14. Grading refinements and the small grading group
  15. Grading refinement data in general
  16. ...and 20 more sections

Key Result

Lemma 2.4

A weighted $A_\infty$-algebra over $R$ is specified by an $R$-bimodule $A$ and maps $\mu_n^w \mathpunct{}\nonscript \mkern-: muplus1mu A^{\otimes_R n}\to A$ for $n,w\in\mathbb Z_{\geq 0}$ such that:

Figures (20)

  • Figure 1: The torus. Left: the torus $T^2$ with $\alpha_1$ and $\alpha_2$ drawn on it and the corners $1,\dots,4$ labeled. Center: the same torus, as an identification space of the rectangle $S$. Right: the pointed matched circle for the torus and labeling of the points, matching, and indecomposable chords.
  • Figure 2: Conventions on a valid labeling. The labels are in $\{1,\dots,4\}$ (mod $4$).
  • Figure 3: Tiling patterns. When these are extended (as in (b) and (c) above), we indicate the distinguished sectors by writing the labels in those sectors. The root is indicated by the large dot on the boundary.
  • Figure 4: Composite labelling convention. We require that $\Lambda$ changes as shown, as we cross $S$ between the two components $\Gamma_1$ and $\Gamma_2$.
  • Figure 5: Example of a composite pattern. Here, $\Gamma_1$ and $\Gamma_2$ are two tiling patterns; we can form their composition $\Gamma_1 \mathbin{\#}_2 \Gamma_2$ to obtain the composite pattern $\Gamma$ on the right. The red dot is the root vertex of $\Gamma_2$, which is to be joined to the $2^{\textrm{nd}}$ boundary arc of $\Gamma_1$.
  • ...and 15 more figures

Theorems & Definitions (167)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.4
  • proof
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Definition 2.9
  • Remark 2.10
  • Remark 2.11
  • ...and 157 more