Pinwheels in symplectic rational and ruled surfaces and non-squeezing of rational homology balls

2025/03/20 by Nikolas Adaloglou, Adaloglou, Nikolas, Johannes Hauber +1
Engineering · Mathematics · #53Dxx #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2503.16250

openalex publication_date 2025/03/20 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

We use almost toric fibrations and the symplectic rational blow-up to determine when certain Lagrangian pinwheels, which we call liminal, embed in symplectic rational and ruled surfaces. The case of L_2,1-pinwheels, namely Lagrangian ℝP²'s, answers a question of Kronheimer in the negative, exhibiting a symplectic non-spin 4-manifold that does not carry a Lagrangian ℝP². In addition, we provide applications to symplectic embeddings of rational homology balls. In particular, we generalize Gromov's classical non-squeezing theorem by proving that a rational homology ball B_n,1(1) embeds into the rational homology cylinder B_n,1(α,∞) if and only if α≥ 1. Along the way, we prove various properties of Lagrangian pinwheels of independent interest, such as describing their homological complement, providing a short proof that performing a symplectic rational blow-up of a Lagrangian pinwheel in a positive symplectic rational manifold yields a symplectic manifold which is also rational, and showing a self-intersection formula for Lagrangian pinwheels.

Related

• Embeddings and disjunction of Lagrangian pinwheels via rational blow-ups
• Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls
• Symplectic Rational Blow-up
• The symplectic topology of some rational homology balls
• Bounds on Embeddings of Rational Homology Balls in Symplectic 4-manifolds