Maps between circle bundles: Fiber-preserving, Finiteness and Realization of mapping degree sets

Let $E_i$ be an oriented circle bundle over a closed oriented $n$-manifold $M_i$ with $i=1,2$ and assume that $M_2$ is aspherical. We prove the following geometric realization result, which extends the classical result for maps between circle bundles over surfaces to all dimensions $\geq 2$. Let $f\