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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.

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    $\begingroup$ It would help if you explain what kind of tools you have available. $\endgroup$
    – Pedro Tamaroff
    Jan 15 '18 at 18:02
  • $\begingroup$ What would be useful is an example of closed path, say in spherical coordinates, that cannot be shrunk to a point. $\endgroup$ Jan 16 '18 at 12:59
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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.

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    $\begingroup$ Thanks for the quick answer, but did you read the part about me being a physicist? I.e. that really doesn't help me, unfortunately. $\endgroup$ Jan 15 '18 at 17:23
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    $\begingroup$ You can't expect us to know what you know about algebraic topology just from the phrase "I am a physicist" (since we are not physicists, and in any case different physicists know different amounts of algebraic topology); if you want us to take your actual background into consideration then describe your actual background. $\endgroup$ Jan 15 '18 at 20:39

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