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 ₁ (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\ \k 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.