Pinwheels in symplectic rational and ruled surfaces and non-squeezing of rational homology balls
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866909545799876608 |
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| author | Adaloglou, Nikolas Hauber, Johannes |
| author_facet | Adaloglou, Nikolas Hauber, Johannes |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16250 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pinwheels in symplectic rational and ruled surfaces and non-squeezing of rational homology balls Adaloglou, Nikolas Hauber, Johannes Symplectic Geometry Algebraic Geometry Geometric Topology 53Dxx 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. |
| title | Pinwheels in symplectic rational and ruled surfaces and non-squeezing of rational homology balls |
| topic | Symplectic Geometry Algebraic Geometry Geometric Topology 53Dxx |
| url | https://arxiv.org/abs/2503.16250 |