Structure of Torus Fibration Under the First Betti Number Restriction
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910211096182784 |
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| author | Peng, Xin Wang, Bing Wang, Zhenjian |
| author_facet | Peng, Xin Wang, Bing Wang, Zhenjian |
| contents | We study torus bundles with affine structure groups. First, we establish a rigidity result under constraints on the first Betti numbers: If $ \text{b}_{1}(M)-\text{b}_{1}(N)=\dim M-\dim N $ holds for a torus bundle $M$ with an affine structure group over a closed manifold $N$, then $M$ can be classified. Second, we obtain some necessary and sufficient conditions for the topological splitting of principal torus bundles. These results improve the understanding of the geometry of collapsing sequences under the first Betti number constraints, thereby extending the prior work by Huang-Wang. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11552 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structure of Torus Fibration Under the First Betti Number Restriction Peng, Xin Wang, Bing Wang, Zhenjian Differential Geometry 53Cxx, 55Rxx We study torus bundles with affine structure groups. First, we establish a rigidity result under constraints on the first Betti numbers: If $ \text{b}_{1}(M)-\text{b}_{1}(N)=\dim M-\dim N $ holds for a torus bundle $M$ with an affine structure group over a closed manifold $N$, then $M$ can be classified. Second, we obtain some necessary and sufficient conditions for the topological splitting of principal torus bundles. These results improve the understanding of the geometry of collapsing sequences under the first Betti number constraints, thereby extending the prior work by Huang-Wang. |
| title | Structure of Torus Fibration Under the First Betti Number Restriction |
| topic | Differential Geometry 53Cxx, 55Rxx |
| url | https://arxiv.org/abs/2605.11552 |