The object of study are topological embeddings $h\colon S^{n-1}\rightarrow S^n$, $n\ge2$. We have the
Jordan-Brouwer separation theorem: If $h\colon S^{n-1}\rightarrow S^n$ is an embedding, then $S^n-h(S^{n-1})$ has exactly two (path-)components. The components are both acyclic and have topological boundary $h(S^{n-1})$.
This theorem is easily accessible using algebraic topology. In case $h$ is the standard equatorial embedding, the closure of each component of $S^n-h(S^{n-1})$ is a disk. It is known that this is not the case in general (the Alexander horned sphere is the historical counterexample). The question of how pathologies can be avoided is answered by the
Generalized Schoenflies theorem: If $h\colon S^{n-1}\rightarrow S^n$ is an embedding and $A$ is the closure of a component of $S^n-h(S^{n-1})$, TFAE:
- The embedding $h$ extends to an embedding $\tilde{h}\colon S^{n-1}\times[0,1]\rightarrow A$.
- $A$ is a topological manifold with boundary.
- $A\cong D^n$.
The implication 3.=>2. is trivial, the implication 2.=>1. a consequence of the existence of collar neighborhoods and 1.=>3. is the heart of the theorem. There also is the classical
Schoenflies theorem: If $h\colon S^1\rightarrow S^2$ is an embedding and $A$ is the closure of a component of $S^2-h(S^1)$, then $A\cong D^2$.
A popular alternative is the (I'm not entirely confident that I'm getting the naming conventions right here)
Jordan-Schoenflies theorem: If $h\colon S^1\rightarrow S^2$ is an embedding, there is a homeomorphism $H\colon S^2\rightarrow S^2$ extending $h$.
It is not hard to see that the last two theorems are equivalent. These are also often phrased in terms of embeddings $S^1\rightarrow\mathbb{R}^2$ rather than $S^1\rightarrow S^2$, but translating between the corresponding versions is easy. My question is: Do these classical theorems, as the name suggests, actually follow from the generalized Schoenflies theorem? That is, is there an easy argument that every embedding $S^1\rightarrow S^2$ is automatically (bi-)collared?
Best Answer
No, there's nothing easier than just going through the proof of the original Schönflies Theorem.
I don't think there's any confusion here, everything is out in the open. The naive "generalization" of the Schönflies Theorem is false in higher dimensions. The weaker "generalization" - i.e. the one where the bicollared condition is incorporated as a hypothesis - is true. The "generalization" terminology is useful here but it should not be over-interpreted.