Connected components of space of maps between two manifolds - MathOverflow

Connected components of space of maps between two manifolds

2010-05-01T20:27:10Z

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.

Answer by Dylan Thurston for Connected components of space of maps between two manifolds

2010-05-02T03:27:50Z

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.

Answer by Dan Ramras for Connected components of space of maps between two manifolds

2010-05-02T06:54:05Z

This got too long for a comment to Dylan's answer.

I like the discussion of these ideas in John Lee's book Introduction to Differentiable manifolds (the relevant part isn't in the google preview). He refers to these approximation results as the Whitney Approximation Theorem, and deduces them from the tubular neighborhood theorem and the Whitney Embedding Theorem. Interestingly, he doesn't use convolution.

Here's a sketch: start with a continuous map from $f:M\to N$ , and embed N in R^n. First Lee proves that there's smooth map $g: M\to R^n$ close to f (inside a tubular neighborhood of N, say) and then he uses the projection from the tubular neighborhood back to N to get his approximation to f. Note that since balls in $R^n$ are convex, once g is sufficiently close to f there's a linear homotopy linking them, which lies entirely inside the tubular neighborhood.

To produce g, the rough idea is that near any point x in M, the constant function with value f(x) is a "good enough" approximation to f. This gives an open cover of M, and there's a finite subcover, with a subordinate partition of unity $\phi_i$ . The approximation to f is now gotten by averaging these constant functions using the partition of unity: $g(x) = \sum \phi_i (x) f(x_i)$ .

Lee explains how to modify this in the case where f is already smooth on some closed subset, and you want to leave it unchanged there. (That allows you to approximate homotopies.)

Answer by Ady for Connected components of space of maps between two manifolds

2010-05-04T21:33:53Z

For the Sobolev case, I think http://www.lincei.it/pubblicazioni/rendicontiFMN/rol/pdf/M2003-03-14.pdf may be helpful [together with the references therein].