The Fox property for codimension one embeddings of products of three spheres into spheres

Abstract

Fox has shown that for every closed connected surface smoothly embedded in $S^3$, the closure of each component of its complement is diffeomorphic to the closure of the complement of a handlebody embedded in $S^3$. In this paper, we study a similar “Fox property” for smooth embeddings of $S^p\times S^q\times S^r$ in $S^{p+q+r+1}$.