# Given a CW-structure on a space, when can a subspace be realized as the n-skeleton?

So I've been working with CW-complexes in my algebraic topology classes for about a year now, and while I certainly feel that I understand them pretty well at this point, there have been some questions about the finer details concerning them that I haven't had the time to sit down and really think about. One of these questions is this:

Suppose that Y is a topological space that is homeomorphic to a CW-complex X and that A is a subspace of Y that is homeomorphic to the n-skeleton of X. When is it true that there exists a homeomorphism from Y to X such that it takes A to the n-skeleton of X? If such a homeomorphism always exists, why is this? If it is not always true that one exists, what's an explicit counter example of this?

Now, I realize that this question might turn out to be trivial, but I think it's also one that I should know the answer to if I plan on continuing studying this subject in the future. Besides just natural curiosity, part of the reason I ask about this is because with many of the canonical examples of cell complexes, this condition does in fact hold true and we can use that fact to deduce things about the (co)homology groups of the pair (Y,A) from cellular (co)homology (e.g. real and complex projective space). Whatever the case is, any answers on the matter would be greatly appreciated.

P.S. If the answer to this is truly obvious I apologize - sometimes I can be a bit of a scatter brain.

Edit: As studiosus pointed out, wild spheres such as the Alexander Horned Sphere, are examples of the kind that I was looking for. However, I would also like to know if there are conditions under which we can say that a subspace can be realized as the n-skeleton of a cw structure on the whole space. If anyone has any information on that, I would also like to know about it. Thank you!

• It does not always exist. Consider a wild 2-sphere in the 3-sphere. There is no CW complex structure on $S^3$ with a wild sphere equal the 2-skeleton. – Moishe Kohan May 13 '16 at 14:52
• @studiosus I'm not familiar with wild spheres. Just to confirm, you are saying that the wild 2-sphere is homeomorphic to the 2-skeleton of the 3-sphere but cannot be realized as its 2-skeleton? Edit: Oh you mean something like the Alexander Horned Sphere? Despite my profile picture, I've actually only worked with it a little bit haha. – Michael Ferguson May 13 '16 at 14:54
• So then maybe this is analogous to the failure of Schonflies Conjecture in higher dimensions? That would actually make a lot of sense. I'm going to make a slight edit to my question then. – Michael Ferguson May 13 '16 at 15:03
• You will probably like Nicholson's theorem and the article itself. – user98602 May 13 '16 at 19:43
• Thank you Mike, this looks very interesting! I look forward to reading it once my exams finish! – Michael Ferguson May 14 '16 at 16:39