# About generalized Schoenflies problem in the smooth category

Reading some books and comparing with Wikipedia I found some different statements about how the smooth Schoenflies problem is solved in high dimension, and I wanted to know which one is the correct one (or maybe if I misunderstood something).

On the book by R. Kirby "The topology of $4$-manifolds" pag. 18 I found this:

[...] This raises the question of the piecewise linear (PL) and smooth Schoenflies Conjectures: A smooth (PL) imbedding of $S^{n-1}$ in $S^{n}$ bounds two smooth (PL) $n$-balls. The PL version is true in dimension other than $4$. The smooth version fails in higher dimensions because of exotic smooth structures on spheres.

On Wikipedia there is the following statement:

The Schoenflies problem can be posed in categories other than the topologically locally flat category, i.e. does a smoothly (piecewise-linearly) embedded $(n − 1)$-sphere in the $n$-sphere bound a smooth (piecewise-linear) $n$-ball? For $n = 4$, the problem is still open for both categories. See Mazur manifold. For $n \geq 5$ the question has an affirmative answer, and follows from the h-cobordism theorem.

Is it true that a smoothly embedded $(n-1)$-sphere divide $S^{n}$ in two smooth $n$-balls? And a related question is: does an exotic $n-1$-sphere bounds a standard $n$-ball?

• Yes, smooth Schoenflies holds in all dimensions $\ne 4$ (open for $n=4$). This indeed follows immediately from the smooth h-cobordism theorem. Thus, Kirby is wrong and Wikipedia is right, how strange... Jun 12, 2014 at 22:05

Here is the proof of Schoenflies conjecture for $n\ge 5$ following the Wikipedia article: It is probably in one of Smale's papers but I did not check.
Let $S\subset S^n$ be a smooth closed submanifold diffeomorphic to $S^{n-1}$. By Jordan's separation theorem, $S$ separates $S^n$ in two components $D_+, D_-$. I will prove that the closure of $D=D_+$ is diffeomorphic to the closed round $n$-ball $B^n$. Indeed, let $C\subset D$ be a small round open ball (whose closure is disjoint from $S$. Then $W=D\setminus C$ is an h-cobordism between $S^{n-1}\cong \Sigma=\partial C$ and $S$. (The proof of this is elementary algebraic topology: First verify that $W$ is simply-connected and then compute homology.) Therefore, by Smale's theorem, $W$ is diffeomorphic to $S^{n-1}\times [0,1]$ with diffeomorphism sending $\Sigma$ to $A=S^{-1}\times 0$ and $S$ to $S^{n-1}\times 1$. Thus, $cl(D)$ is diffeomorphic to the annulus with a round ball attached via a diffeomorphism. To see that the result is diffeo to the round ball just note that every self diffeomorphism of A extends to a self diffeomorphism of its product with the interval.
• The h-cobordism theorem in the smooth category holds for n-dimensional h-cobordisms where $n\geq 6$. So the above proof also applies only to $n\geq 6$. For $n=5$ we need an extra trick. See page 112 of Lectures on the h-cobordism theorem maths.ed.ac.uk/~v1ranick/surgery/hcobord.pdf The trick there is to use the Poincaré conjecture in dimension 5 in the smooth conjecture and the result by Palais that any two smooth maps of a disc into a smooth manifold are isotopic. Jan 29, 2019 at 10:45