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Selective symplectic homology with applications to contact non-squeezing
- Part of
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- Journal:
- Compositio Mathematica / Volume 159 / Issue 11 / November 2023
- Published online by Cambridge University Press:
- 18 September 2023, pp. 2458-2482
- Print publication:
- November 2023
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- Article
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We prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with large symplectic homology: there exists a smoothly embedded ball in such a sphere that cannot be made arbitrarily small by a contact isotopy. These homotopy spheres include examples that are diffeomorphic to standard spheres and whose contact structures are homotopic to standard contact structures. As the main tool, we construct a new version of symplectic homology, called selective symplectic homology, that is associated to a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of Floer homology groups for Hamiltonians whose slopes tend to
$+\infty$ on the open subset but remain close to 
$0$ and positive on the rest of the boundary.
