I need to find the universal covering space of $X=D^2 / \sim$ where $x \sim y$ iff $x=y$ or $ x,y \in S^1$ and $q(x)=q(y)$. Here $q: S^1 \mapsto S^1$ is n sheeted covering space of the circle.
This is what I am thinking: for n=2, this is the disk with antipodal points identified and we know that in this case the universal covering of the real projective plane is $S^2$. I know how to prove this when we consider the real projective plane as the sphere with antipodal points identified (rather than disk with antipodal points identified). Also, in this case $q$ is 2-sheeted covering space and our universal covering $p$ is 2 sheeted (which makes sense because $\pi_1(RP_2)=Z_2$).
Now, for our $X$, it is not hard to prove that $\pi_1(X)=Z_n$. So, I think our universal covering space is going to be n sheeted covering space. But I am having trouble to see what it could be. Is it also a sphere $S^2$ with some particular properties? And if so, how to prove it?
Best Answer
This is written for 3-sheeted covering of circle. But this is generalizable.
Take three discs and identify them along their boundary and consider the following quotient map where same coloured regions are identified. This will give you the universal covering.
It is easy to see that this space is simply connected and the cover is 3-fold.