Maps between circle bundles: Fiber-preserving, Finiteness and Realization of mapping degree sets
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2025
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| _version_ | 1866910961713020928 |
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| author | Neofytidis, Christoforos Sun, Hongbin Tian, Ye Wang, Shicheng Wang, Zhongzi |
| author_facet | Neofytidis, Christoforos Sun, Hongbin Tian, Ye Wang, Shicheng Wang, Zhongzi |
| contents | 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 $n= 4, 5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maps between circle bundles: Fiber-preserving, Finiteness and Realization of mapping degree sets Neofytidis, Christoforos Sun, Hongbin Tian, Ye Wang, Shicheng Wang, Zhongzi Geometric Topology Algebraic Topology Group Theory Number Theory 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 $n= 4, 5$. |
| title | Maps between circle bundles: Fiber-preserving, Finiteness and Realization of mapping degree sets |
| topic | Geometric Topology Algebraic Topology Group Theory Number Theory |
| url | https://arxiv.org/abs/2505.16285 |