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Math Help - Topology Question

  1. #1
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    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!
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  2. #2
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    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.
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