Are all embeddings of $D^n$ into $R^n$ ambiently homeomorphic?

Question

To make the question more precise: let $i, j : D^n \to R^n$ be topological embeddings. Does there exist a topological automorphism $h : R^n \to R^n$ such that $h \circ i = j$?

Answer 1

• Yes if $n=1$ (easy).
• Yes if $n=2$ (not easy: this follows from the Jordan–Schönflies theorem)
• No if $n\ge 3$. The classical counterexample is the Alexander horned sphere. When filled in, it is homeomorphic to closed $3$-ball. However, the exterior is not simply connected and therefore is not homeomorphic to the exterior of the standard $3$-ball.