Let m, n > 1. Is it true that C(S^m, S^m), and C(S^n, S^n) are homeomorphic ? [both endowed with the sup metric (or equivalently the compact-open topology)]

Generally, C(S^n, S^n), with n >= 1, is a countably infinite (disjoint) union of path-connected (due to Hopf) components C_{n,k}, k in Z.

I think each of these components may be viewed as an infinite dimensional Frechet manifold. Unfortunately, they are not contractible. However, the question is [somewhat vaguely] motivated by the Henderson's Theorem.

Also, I have some related questions : - Fixing n, are the C_{n,k}'s homeomorphic to each other (at least for k <> 0) ? - Are there some m, n, k, l with m <> n s.t. C_{m,k} ~ C_{n,l} ?

What I was able to do in this direction until now is to show the existence of a proper, one-to-one, degree-preserving map from C(S^m, S^m) into C(S^n, S^n). Even for m >> n, but far to be surjective.

allcontinuous self-maps of S^m. $\endgroup$ – Ady Dec 13 '09 at 3:10andedit your question to improve the exposition. $\endgroup$ – Scott Morrison♦ Dec 13 '09 at 18:47