We say that a topologically embedded 3–sphere in a smoothing of
Euclidean 4–space is a barrier provided, roughly, no diffeomorphism
of the 4–manifold moves the 3–sphere off itself. In this
paper we construct infinitely many one parameter families of distinct
smoothings of 4–space with barrier 3–spheres. The existence of
barriers implies, amongst other things, that the isometry group of these
manifolds, in any smooth metric, is finite. In particular, S1
can not act smoothly and effectively on any smoothing of 4–space
with barrier 3–spheres.
