**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:** 


1. 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.
2. 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 $A\in H\_{\dim M}(N)$.
3. 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.



Connected components of space of maps between two manifolds
Any continuous map from *M* to *N* is homotopic to a smooth map, and if two smooth maps are homotopic, then they are also smoothly homotopic. (More generally, two homotopic functions are homotopic through a homotopy that is smooth except at the endpoints.) The proof involves convolving with Gaussians, and is standard; I think you can find it in Milnor's *Topology from a Differentiable Viewpoint*, for instance. (It's also appeard on mathoverflow before, but I couldn't find it just now.) The hard issues for smooth vs. continuous functions arise only once you start demanding the maps be injective.


I can't say more about McDuff and Salomon without more context for the quote.


