TITLE: what is the universal cover for $S_g$?
QUESTION [13 upvotes]: Denote $S_g$ to be orientable, compact closed surface with genus $g$; especially, $g=1$, $S_1$ is just the torus, it has the universal cover $\mathbb{R}^2$. 
I heard the universal cover for $S_2$ is upper half plane $\mathbb{H}^2=\{(x,y)\in \mathbb{R}^2|y>0\}$, how to prove this? And what is the universal cover for $S_g$ with $g>2$?

REPLY [9 votes]: 1) Any manifold admit a universal cover and the dimension is the same ;
2) The only two simply connected surfaces are the plane and the sphere.
If you look at your compact surface of genus $2$ as a complex manifold, then there is the famous Uniformization Theorem which states that there are only 3 different simply connected complex manifold of dimension 1 : the plane, the sphere and the half-plane.