arXiv2026
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 $f\colon E_1\to E_2$ be a map. If the induced homomorphism $f_*$ on $π_1$ sends the fiber subgroup of $π_1(E_1)$ to the fiber subgroup of $π_1(E_2)$, then there is a fiber-preserving map in the homotopy class of $f$. Using the above result and a computation of the mapping degree set of fiber-preserving maps between $E_1$ and $E_2$, we obtain the following results on the finiteness and the realization problems of mapping degree sets. ($\mathcal F$) The mapping degree set $D(E_1, E_2)$ is finite if $M_2$ is hyperbolic and the Euler class of $E_2$ is not torsion. ($\mathcal R$) For any finite set $A$ of integers containing $0$ and any $n>2$, $A$ is the mapping degree set $D(M,N)$ for some closed oriented $n$-manifolds $M$ and $N$. Item ($\mathcal F$) extends the previously known $3$-dimensional case to all dimensions $\geq 3$. Item ($\mathcal R$) gives a complete answer to the realization problem on finite sets of integers (containing 0) in all dimensions, including in particular the previously unknown cases in dimensions $ 4$ and $5$.