Realizing crosscap transpositions as monodromies of singular fibrations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914577286955008 |
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| author | Hayano, Kenta |
| author_facet | Hayano, Kenta |
| contents | We introduce a new type of singularity for smooth maps from $4$-manifolds to surfaces, called an $M$-singularity, whose critical locus is a circle contained in a single fiber. We show that the monodromy around an $M$-singularity is a crosscap transposition in the mapping class group of a non-orientable surface. We also introduce $M$-fibrations, namely smooth maps whose singularities consist only of $M$-singularities, and prove that relations among crosscap transpositions give rise to such fibrations on non-orientable $4$-manifolds. We then study handle decompositions associated with $M$-fibrations and their orientation double coverings. In particular, we describe the attaching circles and framings of the two $2$-handles arising from the orientation double cover of an $M$-singularity. Using this description, we construct a closed non-orientable $4$-manifold which admits an $M$-fibration but admits no Lefschetz fibration. We further discuss singularity-theoretic properties of the local model of an $M$-singularity, namely its infinite $\mathcal{A}_e$-codimension and an explicit stable perturbation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_17874 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Realizing crosscap transpositions as monodromies of singular fibrations Hayano, Kenta Geometric Topology 57R45 (Primary) 57K20, 57K40, 58K10 (Secondary) We introduce a new type of singularity for smooth maps from $4$-manifolds to surfaces, called an $M$-singularity, whose critical locus is a circle contained in a single fiber. We show that the monodromy around an $M$-singularity is a crosscap transposition in the mapping class group of a non-orientable surface. We also introduce $M$-fibrations, namely smooth maps whose singularities consist only of $M$-singularities, and prove that relations among crosscap transpositions give rise to such fibrations on non-orientable $4$-manifolds. We then study handle decompositions associated with $M$-fibrations and their orientation double coverings. In particular, we describe the attaching circles and framings of the two $2$-handles arising from the orientation double cover of an $M$-singularity. Using this description, we construct a closed non-orientable $4$-manifold which admits an $M$-fibration but admits no Lefschetz fibration. We further discuss singularity-theoretic properties of the local model of an $M$-singularity, namely its infinite $\mathcal{A}_e$-codimension and an explicit stable perturbation. |
| title | Realizing crosscap transpositions as monodromies of singular fibrations |
| topic | Geometric Topology 57R45 (Primary) 57K20, 57K40, 58K10 (Secondary) |
| url | https://arxiv.org/abs/2605.17874 |