# the fundamental group of $X$ is the symmetric group $S_3$, then whether it has a universal cover?

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.

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.
• we have supposed that $X$ is path connected. – Mathillda Nov 28 '18 at 16:58