Universal cover of a torus "pillow" 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?
 A: 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. 
A: To elaborate on @studiosus's response: Rather than thinking of a long chain of spheres, think of a long chain of sausage links. 
We can think of the usual torus $T$ by starting with a cylinder, $S^1\times [0,1]$, and identifying the top and bottom circles; i.e., $T = S^1\times [0,1]/\sim$ where $(x,0)\sim (x,1)$ for every $x\in S^1$. Now, the torus with no hole, $\Sigma$, can be obtained by identifying that glued circle to a single point. I.e., we take $S^1\times [0,1]/\sim_1$, where $(x,0)\sim_1 (y,1)$ for all $x,y\in S^1$.
But let's do this a different way: First identify all the points $(x,0)\in S^1\times [0,1]$ to a point $P$ and all the points $(y,1)\in S^1\times [0,1]$ to a different point $Q$. (Topologically, this is a sphere, but we're thinking of it as a sausage link.) Now identify the points $P$ and $Q$; topologically, this is a sphere with two points identified, but we're thinking of it as attaching the ends of the sausage link to one another. 
If we think of this as a sequence of two equivalence relations we're modding out by, we have $(x,0)\sim_2 (y,0)$ and $(x,1)\sim_2 (y,1)$ for all $x,y\in S^1$, and on this quotient we then identify $P$ and $Q$.  Thus, we've obtained the identification space $S^1\times [0,1]/\sim_1$, so our sausage link is in fact homeomorphic to the torus with no hole, $\Sigma$.
Now, the universal cover is the infinite chain of sausage links, namely,
$$S^1\times\Bbb R/\sim_4\,, \quad\text{where } (x,n)\sim_4 (y,n) \text{ for all }x,y\in S^1 \text{ and } n\in\Bbb Z.$$
