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Victor
Victor
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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=3$$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?

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 $n=3$.) 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?

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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YCor
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Victor
Victor
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Schoenflies problem in PL setting

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 $n=3$.) 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?