Smooth 4-dimensional Schoenflies problem

Importance: Outstanding ✭✭✭✭
Author(s): Alexander, J
Subject: Topology
Recomm. for undergrads: no
Posted by: rybu
on: November 6th, 2009

\begin{problem} Let $M$ be a $3$-dimensional smooth submanifold of $S^4$, $M$ diffeomorphic to $S^3$. By the Jordan-Brouwer separation theorem, $M$ separates $S^4$ into the union of two compact connected $4$-manifolds which share $M$ as a common boundary. The Schoenflies problem asks, are these $4$-manifolds diffeomorphic to $D^4$? ie: is $M$ unknotted? \end{problem}

By the work of Mike Freedman, $M$ separates $S^4$ into two manifolds which are homeomorphic to $D^4$. So the Schoenflies problem is only non-trivial if $D^4$ admits an exotic smooth structure, which is also an open problem. Although $D^4$ could very well have an exotic smooth structure and yet the Schoenflies problem could have a positive answer. ie: although exotic smooth $D^4$'s might exist, perhaps none of them embed in $S^4$?

Martin Scharlemann has results to the effect that the Schoenflies problem is true provided the embeddings are simple enough.

The smooth Poincare conjecture in dimension 4 is related but disjoint from this problem. For example, the Poincare conjecture could be true and $D^4$ could have an exotic smooth structure -- this would amount to saying the monoid of smooth homotopy 4-spheres has some elements with inverses.

The analogous problem in other dimensions is known to be true. Namely, all embeddings of $S^n$ in $S^{n+1}$ are unknotted (bound manifolds diffeomorphic to $D^{n+1}$) provided $n \neq 3$. For $n=1$ this is due to Schoenflies. For $n=2$ it's due to Alexander (see Hatcher's 3-manifolds notes for a modern exposition). For $n \geq 4$ the result follows from the combination of the Mazur-Brown theorem that an embedding of $S^n$ in $S^{n+1}$ bounds a manifold homeomorphic to $D^{n+1}$, plus a consequence of the H-cobordism theorem which states that $D^{n+1}$ has no exotic smooth structures which restrict to the standard smooth structure on the boundary, provided $n \geq 4$.

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*[A] Alexander, J, On the subdivision of space by a polyhedron. Proc. Nat. Acad. Sci. USA 10 (1924) pg 6--8.

[FQ] Freedman, M. Quinn, F. Topology of 4-manifolds. Princeton University Press.

[S1] Scharlemann, M. The four-dimensional Schoenflies conjecture is true for genus two imbeddings. Topology 23 (1984) 211-217.

[S2] Scharlemann, M. Smooth Spheres in R4 with four critical points are standard. Inventiones Math. 79 (1985) 125-141.

[S3] Scharlemann, M. Generalized Property R and the Schoenflies Conjecture, Commentarii Mathematici Helvetici, 83 (2008) 421--449.

[MMEB] Marston Morse and Emilio Baiada, Homotopy and Homology Related to the Schoenflies Problem. The Annals of Mathematics, Second Series, Vol. 58, No. 1 (Jul., 1953), pp. 142-165

[B] Brown, Morton. A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc., vol. 66, pp. 74–76. (1960)

[MAZ] Mazur, Barry, On embeddings of spheres., Bull. Amer. Math. Soc. 65 (1959) 59--65.

[H] Hatcher, A. 3-manifolds notes. []

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.