# S^2 union a line connecting the north and south pole

I have been trying to determine the fundamental group of S^2 union a line connecting the north and south pole by using the Seifert Van Kampen Theorem. But every time I try and pick my two subsets U and V they are either not open or not arcwise connected or their intersection isn't particularly nice.

My guess is that the fundamental group should be $\mathbb{Z}$, but other than my intuition, I can't seem to find a way to show this.

mathwonk
Homework Helper
look at the universal cover, an infinite string of spheres like beads on a long string.

Thanks! So, I look at the group of deck transformations of the universal covering space (the string of beads), which I believe is Z. Then since the group of deck transformations of the universal covering space is isomorphic to the fundamental group, we are done.

Just for concreteness, what is the reason that the universal cover is this string of beads?

If you wanted to use Van Kampen, couldn't you take the "left of the sphere with the string" along with the "right of the sphere with the string"? Each piece will be homotopic to a circle and their overlap will be homotopic to a figure 8. It seems that the amalgamation will identify the two generators and you indeed get the integers, which is the "obvious" answer.

Oh, I was thinking of your bead incorrectly, I see why it is the universal cover, never mind!

mathwonk