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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

G. Fougeron
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  • From $S^n$ to $R^{n+1}$, not to $R^n$! Also, the map of $B^{n+1}$ should be injective. This is a major open problem in the case $n=3$. The answer is nobody knows. – Moishe Kohan Mar 02 '18 at 17:59
  • Thanks that confirms what the English Wikipedia says. If you write it as an answer, I'll accept it. – G. Fougeron Mar 02 '18 at 18:30

1 Answers1

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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 $E^{k+1}$.

Moishe Kohan
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  • If there does exist a counterexample in dimension 4, then $S^4=B_1 \cup B_2$ where $B_i$ are exotic $B^4$. My understanding is that we could possibly (there might be a proof that such a thing exists given 4D Schoenflies is false but I don't know it) have a PL map $S^4\to B^5$ where the link of a vertex of a point is $B_i$ and we do not know how to rule this out. My understanding is that a similar thing can happen in higher dimensions. In other words, if Schoenflies in dim 4 is false it is plausible that there exists non locally flat PL embeddings in dim >4. – PVAL-inactive Mar 03 '18 at 21:57
  • @Moishe Cohen : I asked a follow-up question at https://math.stackexchange.com/questions/2677526/does-nodes-of-a-polygon-and-cell-relationships-determine-the-shape-of-the-polygo ? If you would be se kind as to have a look, I would be very thankful. – G. Fougeron Mar 05 '18 at 15:05
  • @PVAL-inactive: You are right, in higher dimensions one needs to be more careful when stating the result. Incidentally, did you ever consider removing "inactive" from your user name? – Moishe Kohan Mar 05 '18 at 17:35
  • @MoisheCohen For now my profile/name is representative of the person I want to be, rather than the person I am (not sure if I am joking or serious). – PVAL-inactive Mar 05 '18 at 22:52