On nonorientable $4$--manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915804907307008 |
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| author | Baykur, R. İnanç Morgan, Porter |
| author_facet | Baykur, R. İnanç Morgan, Porter |
| contents | We present several structural results on closed, nonorientable, smooth $4$--manifolds, extending analogous results and machinery for the orientable case. We prove the existence of simplified broken Lefschetz fibrations and simplified trisections on nonorientable $4$--manifolds, yielding descriptions of them via factorizations in mapping class groups of nonorientable surfaces. With these tools in hand, we classify low genera simplified broken Lefschetz fibrations on nonorientable $4$--manifolds. We also establish that every closed, smooth $4$--manifold is obtained by surgery along a link of tori in a connected sum of copies of $\mathbb{CP}^2$, $S^1 \times S^3$ and $S^1\widetilde{\times} S^3$. Our proofs make use of topological modifications of singularities, handlebody decompositions, and mapping classes of surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20950 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On nonorientable $4$--manifolds Baykur, R. İnanç Morgan, Porter Geometric Topology 57K40 We present several structural results on closed, nonorientable, smooth $4$--manifolds, extending analogous results and machinery for the orientable case. We prove the existence of simplified broken Lefschetz fibrations and simplified trisections on nonorientable $4$--manifolds, yielding descriptions of them via factorizations in mapping class groups of nonorientable surfaces. With these tools in hand, we classify low genera simplified broken Lefschetz fibrations on nonorientable $4$--manifolds. We also establish that every closed, smooth $4$--manifold is obtained by surgery along a link of tori in a connected sum of copies of $\mathbb{CP}^2$, $S^1 \times S^3$ and $S^1\widetilde{\times} S^3$. Our proofs make use of topological modifications of singularities, handlebody decompositions, and mapping classes of surfaces. |
| title | On nonorientable $4$--manifolds |
| topic | Geometric Topology 57K40 |
| url | https://arxiv.org/abs/2506.20950 |