Immersed Curves and 4-Manifold Invariants (2501.05634v1)

Abstract: For 3-manifolds with torus boundary, the bordered Heegaard Floer invariants of Lipshitz--Ozsv\'ath--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.

• The paper demonstrates that compositions in the Fukaya category of immersed curves align with compositions of morphisms among bordered Floer invariants.
• By relating immersed curves to bordered Floer invariants, the authors derive new obstructions to smooth equivalences between 4-manifolds and surfaces within the 4-ball.
• The study applies these immersed curve techniques to distinguish previously indistinguishable satellite knots and concordances, suggesting applications in the smooth concordance group.

Immersed Curves and 4-Manifold Invariants

The paper "Immersed Curves and 4-Manifold Invariants" by Jesse Cohen and Gary Guth explores novel interpretations and applications of bordered Heegaard Floer invariants for 3-manifolds with torus boundaries. Specifically, the authors provide significant insights into the geometric and algebraic underpinnings of these invariants through the lens of immersed multi-curves and their associated cobordism maps. This work opens up exciting avenues for understanding and distinguishing smooth structures on 4-manifolds and surfaces within the 4-ball.

Heegaard Floer Homology and Its Extensions

Heegaard Floer homology, initially introduced by Ozsváth and Szabó, is a powerful suite of invariants for closed 3-manifolds. It connects via functorial properties to smooth structures through its cobordism maps derived from 4-dimensional cobordisms between 3-manifolds. The bordered Heegaard Floer homology, developed by Lipshitz, Ozsváth, and Thurston, extends this theory to 3-manifolds with boundaries. It accomplishes this by associating algebraic structures (differential graded modules) rather than chain complexes, broadening its applicability, especially in context with parametrized boundaries.

Immersed Curves and Cobordism Maps

Cohen and Guth leverage the geometric interpretations provided by Hanselman, Rasmussen, and Watson, where bordered invariants of 3-manifolds with torus boundaries are interpreted as immersed multi-curves with local systems in the punctured torus. This paper extends these interpretations to understand morphisms between immersed curve invariants, showing a correspondences to certain cobordism maps. Specifically, they relate the composition within the Fukaya category of these immersed curves to the composition of morphisms in the Heegaard Floer setting.

Key Contributions and Results

1. Fukaya Category Composition: The authors demonstrate that compositions in the Fukaya category, realized through counting holomorphic polygons between three sets of curves, align with the compositions of morphisms among bordered Floer invariants.
2. Obstruction to Smooth Equivalences: Utilizing the established framework, new obstructions to smooth equivalences between 4-manifolds with boundary, and between surfaces within the 4-ball, are inferred. The paper provides examples where distinct cobordism maps lead to distinguishing slice disks for knots that were previously indistinguishable by classical invariants.
3. Applications to Satellite Knots and Concordances: By adopting a satellite construction approach, the paper shows how immersed curve techniques can differentiate between concordances and even suggest cases where satellite operations might be injective in the smooth concordance group.
4. Exploration of Exotic Structures: By combining these algebraic-geometric methods with holomorphic polygon counting, the authors explore frameworks where traditional smooth structures become distinguishable — even suggesting new exotic structures and demonstrating the non-equivalence of cobordism through concrete examples.

Speculations on Future Developments

This work sets the stage for further exploration into the combinatorial properties of immersed curves and their potential utility in high-dimensional topology. The geometric insights gained could lead to more precise invariants capable of distinguishing richly complex 4-dimensional phenomena. Further development may lean towards understanding the influence of these results on related structures, such as the derived categories of symplectic geometry, or even contribute to a deeper understanding of knotted and linked surfaces in 4-manifolds.

The overarching implication is that by delving deeper into the relationships between algebraic invariants and geometric representations, enhancement in both the classification and construction of smooth 4-manifolds may be achieved. The paper indicates numerous potential pathways for extending these ideas to broader contexts — alluding to a rich, untapped framework worthy of further collaborative exploration.