Given a complex $Y$ and a map $Y\to S^1$, I am told that the induced map $\pi_1(Y)\to \pi_1(S^1)$ having image of finite index implies that there is a short exact sequence $$1\to H \to\pi_1(Y)\to \pi_1(S^1)\to 1.$$
I'm not sure why this is true, especially since the map to $\pi_1(S^1)$ is not surjective.