Consider the sphere $S^{4}$ as a subset of $\mathbb{R}^{5}$ and consider the action of the group $G$ of homeomorphisms generated by $(x_1, x_2, x_3, x_4, x_5) \rightarrow (-x_2, x_1, -x_4, x_3, x_5)$
The problem asked to show that the action is free on $M=S^{4}\setminus \{(0, 0, 0, 0, \pm 1\}$ (easy), and then to compute the fundamental group of $M/G$. So, $M$ is simply connected, the action by $G$ on each fiber is $\mathbb{Z}_4$, and the action is proper, free, and continuous, so by the quotient manifold theorem and the automorphism structure theorem, the fundamental group of $M/G$ is $\mathbb{Z}_4$. (It would be nice if someone could double check this).
The last part of the problem that I am stuck on however is showing that $S^{4}/G$ is simply connected. Since the action is no longer free in that case, I do not have any structure theorems (that I know of) to attack this problem. I can't imagine what this space might look like, so no Van Kampen's theorem. And now I am out of ideas.
Thanks for any help!
