Repairing embeddings of $3$-cells with monotone maps of $E^{3}$

Abstract:

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