Question: What are the connected components of the familiar spaces of functions between two (let's say compact and smooth, for simplicity) manifolds $M$ and $N$?
Specifically, I'm thinking of the Hölder spaces $\mathcal{C}^{k,\alpha}(M, N)$ and the Sobolev spaces $\mathcal{W}^{k,p}(M, N)$.
Some comments:
For a smooth function $f:M\to N$, it seems clear that, at least, all continuous functions homotopic to $f$ will be connected to it.
This question is inspired by the discussion of $\mathcal{W}^{k,p}(M, N)$ in McDuff-Salamon's book on $J$-holomorphic curves. There it's stated as an offhand remark that the connected components of $\mathcal{W}^{k,p}(M, N)$ (in the case of $M$ oriented & two-dimensional; I'm not sure if this makes a difference) are the completions of the sets {$f:M\to N \text{ smooth}: f_*[M]=A$}, for $A\in H_{\dim M}(N)$.
If the McD-S factoid is true, there should exist sequences of smooth not-all-mutually-homotopic functions which converge in $\mathcal{W}^{k,p}(M, N)$. (This isn't too counterintuitive, since $\mathcal{W}^{k,p}(M, N)$ presumably contains functions which aren't continuous, & so don't themselves have a homotopy class). Can someone give me an example of this phenomenon?
Please feel free to re-tag -- I can't think of anything really appropriate.
