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

TL;DR

The article classifies when Lagrangian pinwheels $L_{p,q}$, especially liminal ones, embed in symplectic rational and ruled four-manifolds by leveraging almost toric fibrations and symplectic rational blow-ups. It proves sharp homological and symplectic inequalities that constrain the ambient form parameters in $S^2 imes S^2$ and $X_1$, and uses these to answer Kronheimer’s question for $L_{2,1}$ (i.e., Lagrangian $ obreak ext{RP}^2$) and to derive a non-squeezing theorem for rational homology balls. The construction side employs ATFs to realize pinwheels in explicit homology classes, while the obstruction side hinges on Li–Li results for embedded spheres after blowing up, yielding precise bounds on the symplectic areas. Collectively, the results reveal rigid embedding phenomena and extend non-squeezing-type rigidity to rational homology balls, with several corollaries about smooth versus symplectic embeddings. The methods forge connections between algebro-geometric degenerations, almost toric geometry, and symplectic embedding theory, enabling concrete computations for higher pinwheels and related rational balls.

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 $\mathbb{R}P^2$'s, answers a question of Kronheimer in the negative, exhibiting a symplectic non-spin $4$-manifold that does not carry a Lagrangian $\mathbb{R}P^2$. 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 $B_{n,1}(α,\infty)$ if and only if $α\geq 1$. 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.

Figures (13)

• Figure 1: For $\alpha <1$ the "quantitative" rational homology ball $B_{n,1}(1)$ does not embed into the rational homology cylinder of "width" $\alpha$.
• Figure 2: For $m \in \mathbb{N}_{>1}$ two almost toric base diagrams related by a mutation at the lower base-node of almost toric base diagram on the left. A calculation, that is easily reproduced, yields all the directions involved on the right hand side.
• Figure 3: The almost toric base diagram $\Delta_{p,q}$.
• Figure 4: The configuration used to compute the self-intersection number of the visible pinwheel.
• Figure 5: Construction of a Lagrangian $\mathbb{R}P^2$-pinwheel in $(X_1,\omega_{h,\mu})$ for $\mu<\frac{h}{2}$.

Theorems & Definitions (91)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Remark
• Theorem E: \ref{['thrm: kronh']}
• definition 1
• definition 2
• remark 2.1
• definition 3