# Math Help - Topology Question

1. ## Topology Question

Hey! Any help will be greatly appreciated!

1. This is a 2 part question. I got the first part.
i)compute $\Pi (S^1VS^2)$
I know how to do this one, the answer is Z, but I don't get how to do part 2 of the problem.

ii) describe all its covering spaces and the corresponding subgroups of $\Pi_1$

Thank you!

2. Subgroups of Z are of the form <n>={k*n | k \in Z}, that is, all the multiples of a single integer n. When n = 0 <0>={0} is the trivial subgroup.
Each subgroup corresponds to a covering of X=S1 V S2, p : E_n -> X, with n sheets, as follows: Divide the unit circle S^1 evenly to n parts, using the n unit roots of 1 ${p_k = e^{ \frac{\sqrt{-1} 2k \pi}{n} }$. Attach a sphere $S^2$ to each $p_k$, the covering map p is defined to be $p(e^{\sqrt{-1}\theta})=e^{\sqrt{-1}n\theta}$ on the circle and the identity map on the sphere sending the n attaching points to the single attaching point of $S^1 V S^2$.
The universal covering space is a straight line with a sphere attached to each of its integer points. That is, a line with infinitely (countable) many spheres attached.