Fix an algebraic integer $x\neq 0$. Is there a closed 3-manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$?
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Covers of a 3-manifold pull back a cohomology class to any algebraic multiple
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