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Lecture notes on Heegaard Floer homology

C. -M. Michael Wong, Sarah Zampa

Abstract

These are the notes for a lecture series on Heegaard Floer homology, given by the first author at the Rényi Institute in January 2023, as part of a special semester titled ``Singularities and Low Dimensional Topology''. Familiarity with Heegaard diagrams and Morse theory is assumed. We first illustrate the relevant algebraic structures via grid homology, and then highlight the geometric rather than combinatorial nature of the general theory. We then define Heegaard Floer homology in the context of Lagrangian Floer homology, and describe some key properties.

Lecture notes on Heegaard Floer homology

Abstract

These are the notes for a lecture series on Heegaard Floer homology, given by the first author at the Rényi Institute in January 2023, as part of a special semester titled ``Singularities and Low Dimensional Topology''. Familiarity with Heegaard diagrams and Morse theory is assumed. We first illustrate the relevant algebraic structures via grid homology, and then highlight the geometric rather than combinatorial nature of the general theory. We then define Heegaard Floer homology in the context of Lagrangian Floer homology, and describe some key properties.
Paper Structure (22 sections, 11 theorems, 77 equations, 16 figures)

This paper contains 22 sections, 11 theorems, 77 equations, 16 figures.

Table of Contents

  1. Grid homology
  2. Definition and $\partial^2 = 0$
  3. Invariance
  4. Maslov and Alexander gradings
  5. Graded isomorphism types
  6. Other flavors
  7. "Naive" Heegaard Floer homology
  8. Definition
  9. Admissibility
  10. $\partial^2 \equiv 0$
  11. Lagrangian Floer homology
  12. Setup
  13. Morse homology: An interlude
  14. The differential
  15. Technicalities and additional structures
  16. ...and 7 more sections

Key Result

Proposition 1.4

$\mathop{\mathrm{\widetilde{\partial}}}\nolimits$ is a boundary map; i.e.,$\mathop{\mathrm{\widetilde{\partial}}}\nolimits\circ\mathop{\mathrm{\widetilde{\partial}}}\nolimits=0$.

Figures (16)

  • Figure 1: A grid diagram of the left-handed trefoil
  • Figure 2: Two generators of $S(\mathbb{G})$ on a diagram $\mathbb{G}$ of the left-handed trefoil
  • Figure 3: Two rectangles with no shared vertex
  • Figure 4: Intersecting rectangles with a shared vertex
  • Figure 5: Grid diagrams of the unknot of different sizes
  • ...and 11 more figures

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Proposition 1.4
  • proof
  • Remark 1.5
  • Definition 1.6
  • Proposition 1.7
  • proof
  • Example 1.8
  • ...and 33 more