Exotic families of embeddings

Abstract

We construct a number of topologically trivial but smoothly non-trivial families of embeddings of 3 3 -manifolds in 4 4 -manifolds. These include embeddings of homology spheres in S 4 S^4 that are not isotopic but have diffeomorphic complements, and families (parameterized by high-dimensional spheres) of embeddings of any 3 3 -manifold that embeds in a blown-up K 3 K3 surface. In each case, the families are constructed so as to be topologically trivial in an appropriate sense. We also illustrate a general technique for converting a non-trivial family of embeddings into a non-trivial family of submanifolds.