Cusp equivalence between smooth embeddings of the 2-sphere in 4-space

Abstract

If the fundamental group of the complement of a smooth embedding f: S^2 \subset R^4 is a cyclic group, the map can be deformed to the standard embedding by a generic one-parameter family with at most cusp singularities. If two smooth embeddings are connected by such a deformation, they will be called cusp equivalent. We will discuss the relation of three equivalences of smooth 2-knots S^2 \subset R^4; cusp equivalence, stable equivalence and weakly stable equivalence.