Smooth Euclidean 4–spaces with few symmetries

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, S can not act smoothly and effectively on any smoothing of 4–space with barrier 3–spheres. AMS Classification 57R55