Selective symplectic homology with applications to contact non-squeezing

TL;DR

This paper introduces selective symplectic homology to prove a contact non-squeezing phenomenon on certain homotopy spheres, showing that some embedded balls cannot be shrunk via contact isotopies.

Contribution

It develops a new version of symplectic homology called selective symplectic homology, enabling novel non-squeezing results in contact geometry.

Findings

Existence of contact non-squeezing on fillable homotopy spheres.

Construction of selective symplectic homology as a new Floer homology variant.

Identification of non-squeezing phenomena in specific contact manifolds.

Abstract

We prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with infinite dimensional 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 infinity on the open subset but remain close to 0 and positive on the rest of the boundary.