Pinwheels in symplectic rational and ruled surfaces and non‐squeezing of rational homology 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 ‐pinwheels, namely Lagrangian , answers a question of Kronheimer in the negative, exhibiting a symplectic non‐spin four‐manifold that does not carry a Lagrangian . 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 embeds into the rational homology cylinder if and only if . Along the way, we prove various properties of Lagrangian pinwheels of independent interest, such as describing their homological complement, providing a short proof that the 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.