What is the status of the Schoenflies problem in the PL category? In other words, given an injective PL map $f:S^{n-1} \hookrightarrow S^n$, is it always PL equivalent to the equatorial inclusion? (I know the problem is still open for $n=4$.) Relatedly, is it true that a codimension one injective map of closed PL manifolds is always locally flat? Also, relatedly, does topological locally flatness imply PL locally flatness in codimension one?
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1$\begingroup$ I believe that the case of $n = 3$ was settled by Alexander. Perhaps you mean $n = 4$? $\endgroup$– Sam NeadCommented Jun 1 at 17:44
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2$\begingroup$ The problem is settled for all $n\ge 5$, mostly by the PL h-cobordism theorem. $\endgroup$– Moishe KohanCommented Jun 1 at 18:03
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$\begingroup$ @Moishe Kohan For applying the PL h-cobordism theorem one needs to prove first that $f(S^{n-1})$ divides $S^n$ into two topological $n$-discs. This could follow from the topological locally flatness of $f$ and the topological Schoenflies theorem. However, why $f$ is topologically locally flat? For example, given a non-trivial PL knot $f\colon S^3\hookrightarrow S^4$ (existence of such is an open problem), it is not completely obvious why its suspension $\Sigma f\colon \Sigma S^3\hookrightarrow \Sigma S^4$ is topologically locally flat. $\endgroup$– VictorCommented Jun 2 at 16:50
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2$\begingroup$ I see what you are asking about: The PL formulation of Schoenflies that I am familiar with, assumes local flatness. Then the standard reference to a proof is the book by Rourke and Sanderson. Without the local flatness assumption, I do not think it is known. $\endgroup$– Moishe KohanCommented Jun 2 at 16:55
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The answer to this question for any $n\geq 4$ is positive iff it is positive for $n=4$. If there is a Schoenfiles PL embedding $f\colon S^3\hookrightarrow S^4$, which is not PL equivalent to the equatorial inclusion, then $\Sigma^{n-4}f\colon S^{n-1}\hookrightarrow S^n$ is not PL locally flat, $n\geq 5$, by induction looking at the map of links $S^{n-2}=L_a(S^{n-1})\hookrightarrow L_{\Sigma^{n-3}f(a)}(S^{n})=S^{n-1}$. (Note that it would still be topologically locally flat, since $f$ is trivial topologically.) Otherwise, any PL embedding $g\colon S^{n-1}\hookrightarrow S^n$ is PL locally flat and trivial by induction, see page 47 in "Introduction to piecewise-linear topology" by Rourke-Sanderson.
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$\begingroup$ Do you have information about the codimension one case? $\endgroup$– shuhaloCommented Sep 19 at 21:08
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$\begingroup$ I am not sure I understand your question. The Schoenflies problem is the codimension one problem. $\endgroup$– VictorCommented Sep 20 at 22:42
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1$\begingroup$ I was referring to the two "relatedly" questions. $\endgroup$– shuhaloCommented Sep 21 at 17:49
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$\begingroup$ The first two questions have a positive answer iff the answer is positive in the ambient dimension four. In other words, if any smooth or, equivalently, PL embedding $S^3 \to S^4$ bounds a standard disc (in dimension four any PL 4-manifold admits a unique smooth structure), then the answer is positive in all dimensions: any PL inclusion $S^{n-1}\hookrightarrow S^n$ is trivial for $n\geq 4$, any PL codimension one injective map $M^{n-1}\hookrightarrow N^n$ is PL locally flat $n\geq 5$ (in ambient dimension 4 it is known to be true). $\endgroup$– VictorCommented Sep 24 at 20:29
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$\begingroup$ For the last question, I think any PL injective map $M^{n-1}\hookrightarrow N^n$ is topologically locally flat regardless it is PL locally flat or not. But in order to have an example of an injective PL map, which is not PL locally flat (but topologically locally flat) one needs a Schoenflies sphere $f\colon S^3\hookrightarrow S^4$ (a PL or smooth embedding which bounds an exotic disc). $\endgroup$– VictorCommented Sep 24 at 20:30