Smooth structures compatible with a given C^1 structures

Question

On a manifold equipped with C^k atlas (with k>0) there is essentially one smooth structure compatible with the atlas. According to Wikipedia, this is a result due to Whitney. This is in stark contrast with a C^0 atlas, where there might exist many smooth structures or none at all.

I was wondering, what is the underlying reason? What makes once-differentiable functions so much better behaved in terms of finding a smooth atlas?

There are many cases where C^1 makes a world of difference - for example, convergence of Fourier series, but maybe there is some geometric explanation?

Also according to Wikipedia, on the long line (not technically a manifold) there are infinitely many smooth structures all compatible with a given C^k structure, so perhaps there is some topology involved...

Answer 1

See my answer to this question.

Consider the space (in some appropriate sense ) of all invertible germs of $C^{\infty}$ maps from $\mathbb R^n$ to itself fixing the origin. This is homotopy equivalent to $GL_n(\mathbb R)$: we can deform the smooth map $f$ to its linear approximation by going through $f_t(x)=\frac{f(tx)}{t}$ as $t$ goes to $0$. The same applies to invertible $C^k$ germs for finite $k>0$, but not to invertible $C^0$ germs. (All of this is equally true for global diffeomorphisms/homeomorphisms $\mathbb R^n\to\mathbb R^n$, too. )