S3/Z2 not simply connected

Question

Can someone direct me towards or give me a proof that $S_3/\mathbb{Z}_2$ is not simply connected? (This is the three-sphere with opposite points being identical). Appreciate if you could keep in mind that I am a physicist when giving a proof.

Answer 1

Its fundamental group is $\mathbb{Z}/2$, the action of $\mathbb{Z}/2$ on $S^3$ is free and proper and $S^3$ is simply connected.

Take two points of $S^3$ that you identify, and a segment $c$ between them its image $p(c)$ by the covering map $S^3\rightarrow S^3/\mathbb{Z}/2$ is a loop, which is not contractible, otherwise you could have been able to lift the homotopy to a point to $S^3$ and this isn not possible.