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Question: Suppose that $X$ is a path-connected space with $\pi_1(X)=S_3$, which is the 3-symmetric group. I just wonder that whether $X$ has a universal cover.

Try: Based on Hatcher, $X$ has a universal cover iff $X$ is path-connected, locally path-connected, and semilocally simply-connected. However, to prove that $X$ meets the latter two conditions is not easy.

I know that $X$ with $\pi_1(X)=S_3$ can be realized by a CW complex and any CW complex meets these three conditions so that has a universal cover. But this is not the way to show that any such $X$ has a universal cover.

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Knowledge of $\pi_1(X)$ and nothing else will not tell you that $X$ is path-connected, locally so, or semilocally simply-connected. Since Hatcher's criteria is an "if and only if" your question has no definite answer as written. The closest you may be able to get is via a CW-approximation which would be correct up to (weak) homotopy, but would not say anything about $X$ itself.

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  • $\begingroup$ we have supposed that $X$ is path connected. $\endgroup$ – Mathillda Nov 28 '18 at 16:58
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    $\begingroup$ Right, but then you need to get locally p-c and semilocally s-c, and all of these properties are independent. $\endgroup$ – Randall Nov 28 '18 at 17:53

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