The nearby Lagrangian conjecture for pinwheels
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913153842937856 |
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| author | Adaloglou, Nikolas Gómez, Gerard Bargalló i Hauber, Johannes |
| author_facet | Adaloglou, Nikolas Gómez, Gerard Bargalló i Hauber, Johannes |
| contents | 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 $τ_{p,q}$. 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 $(p,q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22473 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The nearby Lagrangian conjecture for pinwheels Adaloglou, Nikolas Gómez, Gerard Bargalló i Hauber, Johannes Symplectic Geometry Algebraic Geometry Differential Geometry Geometric Topology 53D35 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 $τ_{p,q}$. 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 $(p,q)$. |
| title | The nearby Lagrangian conjecture for pinwheels |
| topic | Symplectic Geometry Algebraic Geometry Differential Geometry Geometric Topology 53D35 |
| url | https://arxiv.org/abs/2605.22473 |