Smooth 4-dimensional Schoenflies problem

 Importance: Outstanding ✭✭✭✭
 Author(s): Alexander, J
 Subject: Topology
 Keywords: 4-dimensional Schoenflies sphere
 Posted by: rybu on: November 6th, 2009
Problem   Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?

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

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 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 in are unknotted (bound manifolds diffeomorphic to ) provided . For this is due to Schoenflies. For it's due to Alexander (see Hatcher's 3-manifolds notes for a modern exposition). For the result follows from the combination of the Mazur-Brown theorem that an embedding of in bounds a manifold homeomorphic to , plus a consequence of the H-cobordism theorem which states that has no exotic smooth structures which restrict to the standard smooth structure on the boundary, provided .

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.