I am currently trying to compute the universal cover of the punctured torus. I originally thought it would be a lattice, but while computing the fundamental group I saw that the punctured torus is actually homotopy equivalent to the wedge product of two circles.
I know the universal cover of the wedge of two circles is the Cayley graph that looks like a cross everywhere locally, however the punctured torus isn't quite the same as the figure eight, so I'm wondering what exactly the universal cover of the punctured torus looks like.
Thanks for any help.
[Math] Universal Cover of the Punctured Torus
algebraic-topology
Best Answer
Here is a diabolical argument which one can only love or hate:
Put a complex structure on your torus and delete your point.
Since the universal cover of this punctured torus is simply connected but not compact it is holomorphically isomorphic to $\mathbb C$ or to a disk by Riemann's uniformization theorem.
In both cases it is thus homeomorphic to a disk!
[Actually it is also isomorphic to a disk holomorphically, but never mind]