TITLE: Fundamental group of $S^2$ with north and south pole identified
QUESTION [16 upvotes]: Consider the quotient space obtained by identifying the north and south pole of $S^2$. I think the fundamental group should be infinite cyclic, but I do not know how to prove this. 
If it is infinite cyclic, would this and $S^1$ be an example of two spaces which have isomorphic fundamental groups but are not of the same homotopy type?

REPLY [14 votes]: Another way to see this is to use the theory of covering spaces. Consider the subspace $Y$ of $\mathbf R^3$ obtained by placing a copy of $S^2$ centered at $(n, 0, 0)$ for each even integer $n$ (I'll try to remember to add a picture later). The group $\mathbf Z$ acts on this space by translation, and the quotient $\mathbf Z\backslash Y$ is the space in question.

Since $Y$ is simply connected (can you prove this?), it follows from Proposition 1.40c of Hatcher that $\pi_1(\mathbf Z\backslash Y) \cong \mathbf Z$. I don't see a nice way of proving that the spaces aren't homotopy equivalent without using homology, as Chris does.