I was thinking today, what is the universal cover of a torus with the "donut hole" shrunk to a point?
I am certain it must include a sphere, but that can't be enough because of the point at the center of the manifold.
Is it a wedge of two spheres perhaps?
Intuitively something like this seems correct. How should I think about these types of problems in general? Geometrically, what should I look for when searching for a universal cover?
Best Answer
It is the infinite chains of 2-spheres, each pair of consecutive spheres meet in one point, and consecutive spheres are disjoint. The generator of the covering group acts as the shift sending each sphere to the next one.