Fox has shown that for every closed connected surface smoothly embedded in , the closure of each component of its complement is diffeomorphic to the closure of the complement of a handlebody embedded in . In this paper, we study a similar “Fox property” for smooth embeddings of in .
"The Fox property for codimension one embeddings of products of three spheres into spheres." Algebr. Geom. Topol. 11 (5) 3043 - 3064, 2011. https://doi.org/10.2140/agt.2011.11.3043