Jesse recently recruited me as a special correspondent for the goings-on at Math Overflow. Perhaps he’ll eventually let me blog about other things! To begin I’d like to point out a lovely and easy-to-state but not-so-little problem that appeared on MO.
The above problem is perhaps a representative problem in a family of problems that have received little attention by the geometric topology community, which is the issue of low co-dimension embeddings. They are not well understood. This is because these can be rather difficult problems. More than that, there isn’t an edifice — there’s no standard machine to play with.
For example, the generalized Schoenflies problem asks if a smooth embedding of an n-sphere into an (n+1)-sphere bounds a smooth embedded (n+1)-disc. i.e. are there any “knotted” co-dimension 1 spheres? The MO question is about existence of embeddings, the Schoenflies problem is about uniqueness. The answer to the Schoenflies problem is known to be “no” in all spheres, except perhaps the 4-sphere where it’s still an open problem. But the *way* in which the answer is known to be “no” is rather interesting.
In the case that the ambient sphere is the 2-sphere, this is usually known as the “original” Schoenflies theorem, and it’s usually stated that a simple closed curve in the plane bounds a disc. There’s many proofs of this. The big-concept proof would perhaps be to use the Jordan Curve Theorem to say the curve bounds a compact 2-manifold, and then use the classification of 2-manifolds to argue the genus must be zero, and therefore it must be a disc. A far more elementary argument would be to find a linear “height function” which is a Morse function when restricted to the boundary circle. Cutting the bounded region by level-sets separating the critical points of the Morse function decomposes the bounded region into a union of discs. Moreover, the discs are being glued together according to a pattern, and one soon sees that the “gluing instructions” are governed by a tree. The inductive step is to argue that if one glues two discs together along a common arc in their boundary, you get a disc.
In dimension 3, the argument is very similar to the above, but the regions that the Morse function cuts your 3-manifold into are more complicated. There’s a nice proof in Hatcher’s 3-manifolds notes along these lines. This proof is originally due to Alexander. Making similar arguments he also proved that a torus in the 3-sphere must bound a solid torus on one side.
In high dimensions the argument is quite different. A co-dimension one sphere in another sphere cuts the ambient sphere into two manifolds, both of which are contractible. This is a basic argument with the Seifert and Van Kampen theorem, Mayer-Vietoris, Hurewicz and the Whitehead theorem. The boundary of these manifolds is diffeomorphic to a standard sphere (perhaps strangely embedded) so the h-cobordism theorem kicks in and says these manifolds are standard discs. But this argument does not apply in the case the ambient sphere is 4 or of lower dimension. What does apply in all dimensions is the Brown-Mazur theorem, which says our embedded sphere cuts the ambient sphere into two topological balls. Through this lens, the above h-cobordism theorem argument contributes the fact that there is no exotic smooth structure on a disc which restricts to the standard smooth structure on the boundary. In dimension 4, this is an open problem: does the 4-disc admit an exotic smooth structure? And if it does, does the 4-disc with that exotic smooth structure embed in the 4-sphere? That latter question is equivalent to the 4-dimensional Schoenflies problem.
The above two approaches to the Schoenflies problem break down in entertaining ways when you look at the 4-dimensional case. The high-dimensional argument runs into the issue of exotic smooth structures on discs. The low-dimensional argument runs into the problem that when you “slice” a 3-sphere embedded in the 4-sphere you can get very complicated things. So it becomes a rather delicate surgery problem, one that people haven’t had much luck with.
There’s quite a lot of work on manifold embedding problems in the literature. But the main thrust of the literature tends to hover around the differences between embeddings and simpler things: immersions, or continuous maps. And these differences are smallest when the co-dimension is large. The end result is that low co-dimension embeddings have not received much attention. In particular co-dimension 0, 1 and 2 are particularly problematic since the fundamental group plays a prominent role. And in high dimensions, these fundamental groups can be quite intractible beasts.
Getting back to the MO problem, my initial inclination is the answer should be no. Moreover, I think it should be no for relatively mundane reasons — you should be able to choose your open subset of Euclidean space to be the interior of a compact manifold with smooth boundary. Perhaps there are examples in relatively low dimensions, like in 4-dimensional Euclidean space. Even-dimensional oriented manifolds have an intersection pairing on their middle-dimensional homology, given by perturbing the homology classes to be transverse and taking their signed intersection number. But for subsets of Euclidean space, this pairing is always zero, since you can push classes to be disjoint. So the idea would be to find a subset of Euclidean space whose universal cover has a non-trivial pairing on middle-dimensional homology. For example, due to Whitney it’s known that non-orientable surfaces embed in 4-dimensional Euclidean space with normal Euler class any integer of the form: 2×-4, 2x, 2x+4, …, 4-2x where x is the Euler characteristic of the surface. What one would like to do is build a 4-manifold “around” such a surface in a way that the surface lifts to a compact orientable surface in the universal cover of the 4-manifold, having a non-zero self-intersection number.
Ah! This answers the question. I think I’ll post that as a response!