Do spaces with equivalent fundamental group share the same Universal Cover
Just the question above. I have not been able to find a clear answer for this.
I am leaning towards yes but want to be sure.
Solution 1:
$S^1$ has fundamental group $\mathbb Z$, and universal cover $\mathbb R$.
$S^1\times[0,1]$ also has fundamental group $\mathbb Z$, and has universal cover $\mathbb R\times[0,1]$.
Answer to the follow-up: Not all simply connected spaces (these have trivial fundamental group and are its own universal cover) are homotopy-equivalent: $S^2$ and $[0,1]$.