gn.general topology - Are the C(S^n, S^n)'s homeomorphic ?

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.