2025/05/22 by Christoforos Neofytidis, Neofytidis, Christoforos, Hongbin Sun +7 · 1 citation
Engineering · #Advanced Numerical Analysis Techniques
paper · pdf · doi:10.48550/arxiv.2505.16285
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\ ∪\k⋅ deg(f) | k≠ 0, f\colon M1→ M2 with deg(f)≠ 0 such that f^#(e2)=ke1\, where f^# \colon 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: (\mathcal F) The mapping degree set D(E1, E2) is finite if M2 is hyperbolic and e2 is not torsion. (\mathcal 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 (\mathcal F) extend in all dimensions ≥ 3 the previously known 3-dimensional case (i.e., for maps between circle bundles over hyperbolic surfaces). Item (\mathcal 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.