If $X$ is path conneceted, fundamental group of $X$ is finite and there is a covering map $p: X \to S^1$ then $X$ is simply conneceted (1 connected)
Any ideas how to prove this?
2026-03-27 01:42:28.1774575748
Covering mappings
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Let $p: X \rightarrow S^1$ be a covering map. Then the fundamental group of $X$ is a subgroup of the fundamental group of $S^1$, which is $\mathbb Z$ (not trivial as you stated in your comment). You also know that the fundamental group of $X$ is finite, so in fact it has to be the trivial subgroup. Then $X$ is simply-connected.