Thanks Johannes, I think that your argument fulfils my requirement! And yes indeed, the proof of the uniformization theorem of simply connected Riemann surfaces is far from trivial and extremely deep!

Hi Ken, the example you gave, say $X$, is equivalent to the usual smooth structure on $\mathbf{R}$. Indeed the map $\sqrt[3]:X\rightarrow\mathbf{R}$ is a diffeomorphism!

Very nice answer Ryan, thanks a lot.

Thanks Neil for the nice explanation. This cellular approximation theorem is what I was missing!

Yes I know, but say that you put some restrictions on the number of cells in various dimension, don't you think that this will put some restrictions on the kind of groups that you can get?

By $H^*(G,\mathbf{Z})$, I meant of course group cohomology or if you prefer take any contractible space $Y$ on which $G$ acts freely then I meant the Betti cohomology of $Y/G$ namely $H_B^*(Y/G,\mathbf{Z})$.

You are probably right, but unfortunately, unless I'm mistaken, I thought that $k$ was assumed to algebraically closed in Hartshorne's book. Is there some tricky point to address in characteristic $p$ when $k$ is not perfect?

Thanks a lot Georges for Qing Liu's reference. So what is the dualizing sheaf in general for a smooth projective curve defined over an arbitrary field $k$? Is it still the canonical line bundle?