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I am trying to wrap my head around the following problem: I have three objects lined up horizontally, a $2$-sphere, a circle, and another $2$-sphere. It is the wedge sum $S^2 \vee S^1 \vee S^2$. I am trying to find the fundamental group of this space as well as the covering spaces. For the fundamental group, I believe that I can use van Kampen in the following manner:

$$\begin{align*} \pi_1(S^2 \vee S^1 \vee S^2) &= \pi_1(S^2) * \pi_1(S^1 \vee S^2) \\ \pi_1(S^2 \vee S^1 \vee S^2) &= 0 * \pi_1(S^1 \vee S^2) \\ \pi_1(S^2 \vee S^1 \vee S^2) &= 0 * (\pi_1(S^1) * \pi_1(S^2)) \\ \pi_1(S^2 \vee S^1 \vee S^2) &= 0 * \mathbb Z * 0 \end{align*}$$

Does this make sense?

I am still trying to work out how to find the covering spaces.

  • 1
    Let the space be $W$. Then as you have seen Seifert-van Kampen tells us that $\pi_1 (W)\cong \mathbb{Z}$, and more generally $\pi_1(V\vee U)\cong \pi_1(V)*\pi_1(U)$. But this is also intuitively true: Say I had a big loop in our space $W$. We know the fundamental group of the circle is $\mathbb{Z}$, and since any higher dimensional sphere is simply connected, I can just contract the parts of the loop on the $2$-spheres so that the big loop becomes only a loop on the circle. This gives us a correspondence between loops in the space $W$, and loops in the circle.2011-12-21
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    The universal cover will be an infinite line with a string of $S^2$'s attached to it. The deck transformations are generated by motion two steps to the right. Do you see how to get the other covering spaces?2011-12-21
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    Thanks a lot for the help; I really appreciate it. I'm sorry, but I am not seeing how to get the other covering spaces. What is the best way to begin deriving them?2011-12-21

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