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.

  • $\begingroup$ we have supposed that $X$ is path connected. $\endgroup$ – Mathillda Nov 28 '18 at 16:58
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.