Prove that every surface that can be obtained from $S^2, \mathbb{R}P^2$, and the Klein bottle by performing the connecting sum with a finite number of tori has universal cover $\mathbb{R}^2$ (except $S^2$ and $\mathbb{R}P^2$).
Attempt:
These surfaces are connected, and locally simply connected, so they have a universal cover. This universal cover is a 2-manifold because these surfaces are 2-manifolds, so the cover is also locally homeomorphic to $\mathbb{R}^2$.
The Uniformization Theorem says that every simply connected 1-dimensional complex manifold is conformally equivalent (and thus homeomorphic) to the open disk, the complex plane, or $S^2$.
So I'm guessing it's enough to show that the universal cover is not the sphere. One way is to show that it's not compact, but I'm not sure how to. Here's where I'm stuck.
Any suggestions? Are there any mistakes/missing steps? Thanks.