Repairing embeddings of 3-cells with monotone maps of 𝐸³

Abstract

If S 1 {S_1} is a 2-sphere topologically embedded in Euclidean 3-space E 3 {E^3} and S 2 {S_2} is the unit sphere about the origin, then there may not be a homeomorphism of E 3 {E^3} onto itself carrying S 1 {S_1} onto S 2 {S_2} . We show here how to construct a map f of E 3 {E^3} onto itself such that f | S 1 f|{S_1} is a homeomorphism of S 1 {S_1} onto S 2 {S_2} , f ( E 3 − S 1 ) = E 3 − S 2 f({E^3} - {S_1}) = {E^3} - {S_2} and f − 1 ( x ) {f^{ - 1}}(x) is a compact continuum for each point x in E 3 {E^3} . Similar theorems are obtained for 3-cells and disks topologically embedded in E 3 {E^3} .