Timeline for Schoenflies problem in PL setting

10 events

when toggle format	what		by	license	comment
Jun 21 at 16:19	vote	accept	Victor
Jun 20 at 14:37	vote	accept	Victor
Jun 21 at 16:19
Jun 7 at 20:49	answer	added	Victor		timeline score: 1
Jun 2 at 16:55	comment	added	Moishe Kohan		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.
Jun 2 at 16:50	comment	added	Victor		@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.
Jun 2 at 4:02	history	edited	Victor	CC BY-SA 4.0	edited body
Jun 1 at 18:03	comment	added	Moishe Kohan		The problem is settled for all $n\ge 5$, mostly by the PL h-cobordism theorem.
Jun 1 at 17:44	comment	added	Sam Nead		I believe that the case of $n = 3$ was settled by Alexander. Perhaps you mean $n = 4$?
Jun 1 at 16:24	history	edited	YCor		edited tags
Jun 1 at 15:41	history	asked	Victor	CC BY-SA 4.0