I'm having trouble understanding the status of the generalized polygonal Schoenflies problem : The french and the english version in wikipedia seem (at least to me) to state different things, as there might be problems with dimensions $d\le 5$. Moreover, all relevant litterature seem to date from the 1960s.
Here is the english wikipedia page. https://en.wikipedia.org/wiki/Schoenflies_problem#Generalizations
My question is the following : Let us assume there is a piecewise affine injective map from $S^n$ to $\mathbb{R}^{n+1}$. Can it be extended to a injective piecewise affine map from the ball to what should be called the 'interior' of the polygon ?
Thanks
The remaining (major) open problem (as far as Schoenflies is concerned) is the following:
Problem. Let $f: S^3\to R^4$ be a PL embedding. Does $f$ extend to a PL embedding $B^4\to R^4$?
Since in dimensions $\le 6$ PL and DIFF categories are equivalent, you can replace "PL" with "smooth" in this problem. As for which way it will go, it's anybody's guess.
See also this question and this one.
Edit. (Per PVAL's comments): While what I said about dimension 4 is correct, in higher dimensions one has to be more careful in stating the known result ("Alexander-Schoenflies" Theorem in higher dimensions). The precise formulation of that theorem is the following (where $k\ne 3$):
Theorem A (PL category). Let $f: S^k \to E^{k+1}$ be a locally flat PL embedding. Then $f$ is PL isotopic to an embedding whose image is the standard round sphere.
Theorem B (Smooth category). Let $f: S^k \to E^{k+1}$ be a smooth embedding. Then $f$ is smoothly isotopic to a diffeomorphism to the standard round sphere in $E^{k+1}$.