The Embedding of Two-Spheres in the Four-Sphere

We consider the question of reducing the homeomorphism problem for pairs of topological spaces to the homeomorphism problem for single spaces.Of particular interest is the strength of the invariant X -A of the pair (X, A).In the first chapter we discuss the problem in its general setting.In the second chapter we consider the embedding of «-spheres in the «4-2 sphere, and in particular the case « = 2. Instead of the complement Sn+2 -S", we use the exterior of Sn in Sn+2 (i.e., the complement of an open regular neighborhood of Sn).We prove that there are at most two nonequivalent locally flat embeddings of S2 in S4 with homeomorphic exteriors.In certain classical cases, the exterior is a complete invariant for the embedding.In the third chapter we consider the structure of embeddings in more detail.The main result is that any locally flat orientable surface in four-space is the boundary of an orientable three-manifold in four-space.In the fourth and final chapter, we consider the embedding of «-spheres in the « + 2 sphere for values of « other than 2. The situation for w>2 resembles that for n = 2, while the case w=l stands apart from the rest. I. The general problem