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Immersed Curves and 4-Manifold Invariants

Jesse Cohen, Gary Guth

TL;DR

The paper develops a geometric framework that encodes bordered Heegaard Floer invariants for 3-manifolds with torus boundary as immersed curves in the punctured torus and situates them inside the partially wrapped Fukaya category. It proves that composition of bordered invariants aligns with Fukaya composition, so cobordism maps can be computed via triangle counts of immersed curves, thereby providing a concrete bridge between algebraic bordered invariants and symplectic topology. By extending this to satellites, splicing, and Spin$^c$ conjugation, the authors derive practical, curve-based methods to distinguish 4-manifolds and concordances, reinterpreting knot Floer data and involutive structures in terms of immersed curves and their local systems. These results yield new computable obstructions to smooth equivalences in 4-manifolds with boundary and offer a robust toolkit for studying satellites, concordances, and cobordisms within the immersed-curve/Fukaya framework.

Abstract

For 3-manifolds with torus boundary, the bordered Heegaard Floer invariants of Lipshitz--Ozsváth--Thurston have a geometric interpretation as immersed multi-curves with local systems in the punctured torus according to the work of Hanselman--Rasmussen--Watson. We consider morphisms between these immersed curve invariants and show that they compute certain cobordism maps. More precisely, we relate composition in the Fukaya category of immersed curves in the punctured torus to composition of morphisms between the bordered Floer invariants, which have interpretations in terms of certain cobordism maps. We make use of this formalism to obstruct smooth equivalences between 4-manifolds with boundary, and between surfaces with boundary in the 4-ball.

Immersed Curves and 4-Manifold Invariants

TL;DR

The paper develops a geometric framework that encodes bordered Heegaard Floer invariants for 3-manifolds with torus boundary as immersed curves in the punctured torus and situates them inside the partially wrapped Fukaya category. It proves that composition of bordered invariants aligns with Fukaya composition, so cobordism maps can be computed via triangle counts of immersed curves, thereby providing a concrete bridge between algebraic bordered invariants and symplectic topology. By extending this to satellites, splicing, and Spin conjugation, the authors derive practical, curve-based methods to distinguish 4-manifolds and concordances, reinterpreting knot Floer data and involutive structures in terms of immersed curves and their local systems. These results yield new computable obstructions to smooth equivalences in 4-manifolds with boundary and offer a robust toolkit for studying satellites, concordances, and cobordisms within the immersed-curve/Fukaya framework.

Abstract

For 3-manifolds with torus boundary, the bordered Heegaard Floer invariants of Lipshitz--Ozsváth--Thurston have a geometric interpretation as immersed multi-curves with local systems in the punctured torus according to the work of Hanselman--Rasmussen--Watson. We consider morphisms between these immersed curve invariants and show that they compute certain cobordism maps. More precisely, we relate composition in the Fukaya category of immersed curves in the punctured torus to composition of morphisms between the bordered Floer invariants, which have interpretations in terms of certain cobordism maps. We make use of this formalism to obstruct smooth equivalences between 4-manifolds with boundary, and between surfaces with boundary in the 4-ball.
Paper Structure (13 sections, 19 theorems, 90 equations, 24 figures)

This paper contains 13 sections, 19 theorems, 90 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Bordered Floer homology and immersed curves
  4. The partially wrapped Fukaya category of the punctured torus
  5. Immersed Curves and Composition
  6. Composition for 3-manifolds
  7. Composition and Knot Floer Homology
  8. Composition for satellites
  9. An algebraic lemma
  10. Computations and Applications
  11. Satellite concordances via immersed curves
  12. Splicing concordances
  13. $\mathrm{Spin}^{c}$-Conjugation Action

Key Result

Theorem 1

Suppose that $M$ and $N$ are 3-manifolds with parametrized torus boundary. Then, the bordered invariants of $M$ and $N$ can be represented by immersed curves $\bm{\vartheta}_M$ and $\bm{\vartheta}_N$ in $\partial M$ and $\partial N$ respectively. Moreover, if $h: \partial N \rightarrow \partial M$ i

Figures (24)

  • Figure 1.1: The immersed multicurve invariant for $K = m9_{46}$ and the associated knot Floer complex. The map $F_D + F_{D'}$ is given by the small blue triangles.
  • Figure 1.2: A pair of slice disks for $K=m9_{46}$.
  • Figure 1.3: The cobordism maps induced by $1/2$-surgery on the disks $D$ and $D'$.
  • Figure 1.4: The Whitehead double pattern (dark blue) and the two triangles specifying $F_{\mathrm{Wh}(D)}+F_{\mathrm{Wh}(D')}$ (right).
  • Figure 2.1: Conventions for constructing a train track in the torus from a type-$D$ structure.
  • ...and 19 more figures

Theorems & Definitions (46)

  • Theorem 1: Hanselman--Rasmussen--Watson, hanselman2023bordered
  • Remark 1.1
  • Theorem 2
  • Example 1.2
  • Remark 1.3
  • Theorem 3
  • Example 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Definition 2.1
  • ...and 36 more