Contact non-squeezing via selective symplectic homology

I will introduce a new version of symplectic homology that resembles the relative symplectic homology and that is related to the symplectic homology of a Liouville sector. This version, called selective symplectic homology, is associated with a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of the 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. As an application, I will 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 closed 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.