On Embeddings of Spheres

Imbed a n w 1 sphere in an n sphere, and the complement is divided into two components. I t seems that the closure of each of the resulting components should be a topological w-cell. This statement isn't true. The classical counterexample (in dimension 3) is the Alexander Horned Sphere. I t was conjectured, however, that if one restricts one's attention to some class of well-behaved imbeddings, then the statement is true. For instance, in the differentiable case, the Schoenflies Problem asks an even stronger question: Given : S~ —>E, a differentiable imbedding of the (ra —1) -sphere in Euclidean space, can one extend to a differentiable imbedding of the unit ball (of which S~ is the boundary) into Euclidean space? And, in fact, proofs exist for the usual categories of nice imbeddings: differentiable and polyhedral, in dimensions 1, 2, and 3. The problem, then, is to prove this statement for arbitrary dimension N. Such a proof follows under a niceness condition which includes the condition of differentiability.