Let $S^1 = \{z\in\mathbb{C}:|z|=1\}$. For all $n\in\mathbb{N}$, define $f_n: S^1\to S^1$ by $f_n (z) = z^n$. Given $n\in\mathbb{N}$, for what values of $m\in \mathbb{N}$ there exists a lifting of $f_m$ to the recovering $f_n$?
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Recall that we can lift a continuous map $\psi : Y \to X$ to a covering space $\rho : \tilde{X} \to X$ if and only if the map induced on fundamental groups by $\psi$ sends $\pi_{1}(Y,y_0)$ into the image of $\pi_1(\tilde{X},\tilde{x}_0)$ under the map induced by $\rho$, where $\tilde{x}_0$ and $y_0$ are base points such that $\psi(y_0)=\rho(\tilde{x}_0)$.
Since $S^1$ is path connected, we don't really need to pay much attention to the base points, and they can be chosen to work out as above. In this case, all of the fundamental groups involved are isomorphic to the integers $\mathbb{Z}$, so the problem reduces to thinking about subgroups of $\mathbb{Z}$. When is $m\mathbb{Z}$ a subgroup of $n\mathbb{Z}$?
James Staff
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