ON THE ASPHERICITY OF KNOT COMPLEMENTS

Two examples of topological embeddings of S2 in S4 are constructed. The first has the unusual property that the fundamental group of the complement is isomorphic to the integers while the second homotopy group of the complement is non-trivial. The second example is a non-locally flat embedding whose complement exhibits this property locally. Two theorems are proved. The first answers the question of just when good pi1 implies the vanishing of the higher homotopy groups for knot complements in S4. The second theorem characterizes local flatness for 2-spheres in S4 in terms of a local pi1 condition.