Exotic families of embedded spheres

Abstract

We study the difference between the homotopy groups of spaces of smooth embeddings and spaces topological embeddings of a sphere into four-manifolds. In particular, we show that: [see PDF file for equation] may have an arbitrarily high-rank summand for a some 4-manifolds. Here, [see PDF file] represents the component to the embedding space containing a specific embedding S. This behavior is found for spheres of arbitrary self-intersection. We also establish an analogous result for homology groups.