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

TL;DR

The article extends 3-manifold results to higher dimensions by studying maps between oriented $S^1$-bundles over closed aspherical manifolds. It proves that under SCF on the target base, any non-zero degree map is homotopic to a fiber-preserving one, and it derives an explicit formula for the degrees of such maps in terms of base data and cohomology: $D_{FP}(\tilde{M}_a, \tilde{N}_b)=\{0\}\cup\{k\cdot\deg(f)\mid k\neq0, f:M\to N, \deg(f)\neq0, f^{\#}(b)=k a\}$. This enables finiteness results when the base is hyperbolic and the Euler class is non-torsion, and it leads to a complete realization theorem: every finite subset of integers containing $0$ occurs as a mapping degree set $D(M,N)$ for some closed oriented $n$-manifolds $M,N$ in any dimension $n>2$, including new cases in dimensions $4$ and $5$. The work thus connects geometric group properties, circle-bundle topology, and degree theory to address finiteness and realization questions in a broad, higher-dimensional setting.

Abstract

Let $E_i$ be an oriented circle bundle over a closed oriented aspherical $n$-manifold $M_i$ with Euler class $e_i\in H^2(M_i;\mathbb{Z})$, $i=1,2$. We prove the following: (i) If every finite-index subgroup of $π_1(M_2)$ has trivial center, then any non-zero degree map from $E_1$ to $E_2$ is homotopic to a fiber-preserving map. (ii) The mapping degree set of fiber-preserving maps from $E_1$ to $E_2$ is given by $$\{0\} \cup\{k\cdot \mathrm{deg}(f) \ | \, k\ne 0, \ f\colon M_1\to M_2 \, \text{with} \, \mathrm{deg}(f)\ne 0 \ \text{such that}\, f^\#(e_2)=ke_1\},$$ where $f^\# \colon H^2(M_2;\mathbb{Z})\to H^2(M_1;\mathbb{Z})$ 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 $D(E_1, E_2)$ is finite if $M_2$ is hyperbolic and $e_2$ 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 $\geq 3$ 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 $n= 4, 5$.

Theorems & Definitions (38)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4: Finiteness Theorem
• Theorem A
• Theorem B
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Theorem C: Realization Theorem