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Canonical orientations in Heegaard Floer theory

Mohammed Abouzaid, Ciprian Manolescu

TL;DR

The paper develops Heegaard Floer theory over the integers by introducing canonical orientations arising from coupled Spin structures on Lagrangian tori, establishing naturality and functoriality with integral coefficients for HF, sutured HF, and link HF, and providing a new integer-coefficient proof of the surgery exact triangle along with a Z-coefficient version of involutive HF. It extends these constructions to twisted coefficients and to sutured and multi-basepoint variants, yielding a cohesive integral framework across closed, sutured, and link invariants. The work combines Spin/Pin geometry with Perutz’s symplectic perspective to achieve sign-controlled, naturality-rich invariants with potential implications for integral cobordism maps in low-dimensional topology. Overall, the results place Heegaard Floer theory over $\mathbb{Z}$ on solid footing, enabling finer arithmetic and functorial applications in 3- and 4-dimensional topology.

Abstract

We set up Heegaard Floer theory over the integers, using canonical orientations coming from coupled Spin structures on the Lagrangian tori. We prove naturality of Heegaard Floer homology, sutured Floer homology, and link Floer homology over $\mathbb{Z}$. We give a new proof of the surgery exact triangle in this context, as well as a definition of involutive Heegaard Floer homology over $\mathbb{Z}$.

Canonical orientations in Heegaard Floer theory

TL;DR

The paper develops Heegaard Floer theory over the integers by introducing canonical orientations arising from coupled Spin structures on Lagrangian tori, establishing naturality and functoriality with integral coefficients for HF, sutured HF, and link HF, and providing a new integer-coefficient proof of the surgery exact triangle along with a Z-coefficient version of involutive HF. It extends these constructions to twisted coefficients and to sutured and multi-basepoint variants, yielding a cohesive integral framework across closed, sutured, and link invariants. The work combines Spin/Pin geometry with Perutz’s symplectic perspective to achieve sign-controlled, naturality-rich invariants with potential implications for integral cobordism maps in low-dimensional topology. Overall, the results place Heegaard Floer theory over on solid footing, enabling finer arithmetic and functorial applications in 3- and 4-dimensional topology.

Abstract

We set up Heegaard Floer theory over the integers, using canonical orientations coming from coupled Spin structures on the Lagrangian tori. We prove naturality of Heegaard Floer homology, sutured Floer homology, and link Floer homology over . We give a new proof of the surgery exact triangle in this context, as well as a definition of involutive Heegaard Floer homology over .
Paper Structure (35 sections, 42 theorems, 210 equations, 25 figures)

This paper contains 35 sections, 42 theorems, 210 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Spin, Pin, and coupled Spin
  3. Pin structures
  4. Coupled orientations
  5. Coupled Spin structures
  6. Canonical orientations in Lagrangian Floer homology
  7. Orientation spaces
  8. Lagrangian Floer homology
  9. Bubbles
  10. Decompositions
  11. Twisted coefficients
  12. Polygon maps
  13. Examples
  14. Three-manifold invariants
  15. Heegaard Floer homology: a preliminary version
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $Y$ be a closed, oriented $3$-manifold equipped with a basepoint $z \in Y$ and a $\text{Spin}^{\text{c}}$ structure $\mathfrak{s}$. The Heegaard Floer homologies $\widehat{\mathit{HF}}$, ${\mathit{HF}}^+$, ${\mathit{HF}}^-$ and ${\mathit{HF}}^\infty$ (defined using the canonical coupled Spin str

Figures (25)

  • Figure 1: Orientations for gluing a disk bubble to a constant trajectory.
  • Figure 2: Two circles in an annulus.
  • Figure 3: (a) A trivial triangle at a triple intersection. (b), (c): Its perturbations.
  • Figure 4: A handleslide. We draw the $\beta$ circles in blue and the $\gamma$ circles in green. The gray disks are feet of handles that can be connected to other parts of the diagram.
  • Figure 5: An index zero holomorphic triangle contributing $+1$.
  • ...and 20 more figures

Theorems & Definitions (138)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 128 more