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Importance: Outstanding ✭✭✭✭
Author(s): Alexander, J
Subject: Topology
Keywords: 4-dimensional
Schoenflies
sphere
Recomm. for undergrads: no
Posted by: rybu
on: November 6th, 2009
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?

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 $.

Related problems

Smooth 4-dimensional Poincare conjecture
What is the homotopy type of the group of diffeomorphisms of the 4-sphere?

Bibliography

*[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. [http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html]


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