Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces $X_n$ that all cover $X_1 = X$. How do I see that the dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Remark. Of course, the hypothesis of the simplicial map is equivalent to being given a homomorphism $\pi_1X \to \mathbb{Z}$.


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