I know that the universal cover of the plane minus the origin is the plane with the exponential map, but I can't think of the analogue with two points removed. I figured out what the universal cover of two wedged circles looks like (a sort of fractal-like infinite tree thingy in the plane), but I'm not sure if this is relevant (my guess that it might be relevant is that the two are homotopic).
Asked
Active
Viewed 1,264 times
7
-
As the plane minus two points is homeomorphic to a "thick" wedge of two circles, one way to construct its universal covering space is to "thicken" that tree you mention. – Mariano Suárez-Álvarez Jan 29 '18 at 20:11
-
Conformally, this universal cover is the open unit disk. – Moishe Kohan Jan 29 '18 at 20:18
1 Answers
5
The universal cover of the plane minus two points is a plane. The covering map, though, is complicated!
Mariano Suárez-Álvarez
- 135,076
-
2There is nothing special about the two points, really: there are two simply connected manifolds of dimension $2$, the sphere and the plane, and the covering space of your punctured plane has to be one of these. It cannot be the sphere, for your space is not compact. Notice this applies to all connected open subsets of the plane. – Mariano Suárez-Álvarez Jan 29 '18 at 20:14