Abstract:The Lagrangian skeleton of the rational homology ball $B_{p,q}$, for $0<q<p$ coprime integers, is an immersed but not embedded Lagrangian, called a $(p,q)$-pinwheel. We show that any two embeddings of Lagrangian $(p,q)$-pinwheels in $B_{p,q}$ are related by a compactly supported Hamiltonian isotopy, establishing Arnold's nearby Lagrangian conjecture for this wide class of singular Lagrangians. Our proof has two largely independent parts: the first uses neck-stretching and the symplectic rational blow-up to understand embeddings of pinwheels up to symplectomorphism; the second computes that $\text{Symp}_c(B_{p,q})$ is generated by a twist about the pinwheel, which we call the pintwist $\tau_{p,q}$. We provide three applications of our methods: Gromov non-squeezing for pin-balls; a new proof of the local Lagrangian unknotting theorem of Eliashberg--Polterovich; and that the only Lagrangian $(n,m)$-pinwheel in $B_{p,q}$ is of type $(p,q)$.
| Comments: | 68 pages, 23 figures. Comments are very welcome! |
| Subjects: | Symplectic Geometry (math.SG); Algebraic Geometry (math.AG); Differential Geometry (math.DG); Geometric Topology (math.GT) |
| MSC classes: | 53D35 |
| Cite as: | arXiv:2605.22473 [math.SG] |
| (or arXiv:2605.22473v2 [math.SG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.22473 arXiv-issued DOI via DataCite |
From: Nikolas Adaloglou [view email]
[v1]
Thu, 21 May 2026 13:33:44 UTC (417 KB)
[v2]
Fri, 22 May 2026 05:30:15 UTC (403 KB)
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