# Fundamental group of two copies of $S^2$ joined (separately) at their north and south poles

Let $$X$$ be the space obtained by gluing two copies of $$S^2$$ together by identifying their north poles together, and separately their south poles together. Compute $$\pi_1(X)$$.

My thoughts so far: first of all, a quick sketch of the space -

Here, the red points are the north and south poles. I've tried applying van Kampen, but this doesn't seem to work for me since I keep getting intersections which are not path connected. Note that this space can be thought of as the torus where we collapse two distinct circles (one for each red point), does this help? What's a good way to compute $$\pi_1(X)$$ here? Deformation retractions maybe? Hints are appreciated.

• Can you show it is homotopy equivalent to $S^1\vee S^2 \vee S^2$? Jun 4 '21 at 19:53
• I think I can see that, does the $S^1$ come from expanding the red points into line segments that join the north poles together and similarly join the south poles together, and then shrinking the endpoints together to form $S^2\vee S^1\vee S^2$? Not sure if this makes sense... Jun 4 '21 at 20:01
• You can think about starting with the torus, and choosing two circles in it (same angle coordinate). If you collapse one, you get $S^1\vee S^2$, and you can arrange it so that the other circle is in the $S^2$ you get and that $S^1$ passes through it. Then you collapse that circle, and you get what you write. Jun 4 '21 at 20:10

For Van Kampen's Theorem, try letting one of the open sets $$U$$ be a neighborhood of that inner black circle $$C$$, where $$U$$ is chosen so that $$C$$ is a deformation retract of $$U$$.
The set $$X-C$$ is a union of a disjoint pair of open sets $$V_{left}$$ and $$V_{right}$$. Now just do a two step Van Kampen, first with $$U \cup V_{left}$$, and next with $$(U \cup V_{left}) \cup V_{right}$$.