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

Let Eᵢ be an oriented circle bundle over a closed oriented aspherical n-manifold Mᵢ with Euler class eᵢ∈ H²(Mᵢ;Z), $i=1,2$. We prove the following: (i) If every finite-index subgroup of π₁(M₂) has trivial center, then any non-zero degree map from E₁ to E₂ is homotopic to a fiber-preserving map. (ii) The mapping degree set of fiber-preserving maps from E₁ to E₂ is given by \0\ ∪· deg(f) \ | \, k≠ 0, \ f M₁→ M₂ \, with \, deg(f)≠ 0 \ such that\, f^#(e₂)=ke₁\, where f^# H²(M₂;Z)→ H²(M₁;Z) is the induced homomorphism. As applications of (i) and (ii), we obtain the following results with respect to the finiteness and the realization problems for mapping degree sets: (F) The mapping degree set D(E₁, E₂) is finite if M₂ is hyperbolic and e₂ is not torsion. (R) For any finite set A of integers containing $0$ and each $n>2$, A is the mapping degree set $D(M,N)$ for some closed oriented n-manifolds M and N. Items (i) and (F) extend in all dimensions ≥ 3 the previously known $3$-dimensional case (i.e., for maps between circle bundles over hyperbolic surfaces). Item (R) gives a complete answer to the realization problem for finite sets (containing $0$) in any dimension, establishing in particular the previously unknown cases in dimensions $n= 4, 5$.