  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.
