$4D$ analog of Alexander's Horned Sphere

Question

Is it a trivial exercise to find a $4D$ analog to Alexander's Horned Sphere? In other words, is there a manifold homeomorphic to $S^3$, embedded in $\mathbb {R}^4$ that has a wild complement?

Thank you.

Answer 1

I recommend to have a look into

Rushing, T. Benny. Topological embeddings. Vol. 52. Academic Press, 1973.

Theorem 2.6.1 gives a positive answer to your question. But this is not a "trivial exercise".

However, if you have wild sphere $S \subset \mathbb{R}^n$, then you get an obvious embedding of the suspension $\Sigma S$ into $\mathbb{R}^{n+1}$.

If $\mathbb{R}^n \backslash S$ is not uniformly locally 1-connected (1-ULC, see Rushing), then one can show that $\mathbb{R}^{n+1} \backslash \Sigma S$ also fails to be 1-ULC. Thus $\Sigma S \subset \mathbb{R}^{n+1}$ is again a wild sphere. This applies to the horned sphere.