Fiber-preserving and orientation-reversing involutions of Seifert fibered 3-manifolds
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912834078638080 |
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| author | Peet, Benjamin |
| author_facet | Peet, Benjamin |
| contents | We consider fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds and the conditions on a manifold for admissibility of such involutions. We construct a class $Ψ$ of fiber-preserving, orientation-reversing involutions that act trivially on the base. Each element of $Ψ$ is obtained by extending a product involution across Seifert pieces of type $V(2,2;-1)$ - a solid torus with three fibers filled according to Seifert invariants $(2,1)$, $(2,1)$, and $(1,-1)$. We show that $Ψ$ forms a single conjugacy class under fiber-preserving diffeomorphisms. Our main result establishes that any fiber-preserving, orientation-reversing involution factors as $ψ\circ g$, where $g$ is fiber-preserving and orientation-preserving and $ψ\inΨ$, thus reducing the problem to the previously known orientation-preserving case. Through the orientable base-space double covering, we further extend the classification to manifolds with non-orientable base orbifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13469 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fiber-preserving and orientation-reversing involutions of Seifert fibered 3-manifolds Peet, Benjamin Geometric Topology 57S25, 55R05 We consider fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds and the conditions on a manifold for admissibility of such involutions. We construct a class $Ψ$ of fiber-preserving, orientation-reversing involutions that act trivially on the base. Each element of $Ψ$ is obtained by extending a product involution across Seifert pieces of type $V(2,2;-1)$ - a solid torus with three fibers filled according to Seifert invariants $(2,1)$, $(2,1)$, and $(1,-1)$. We show that $Ψ$ forms a single conjugacy class under fiber-preserving diffeomorphisms. Our main result establishes that any fiber-preserving, orientation-reversing involution factors as $ψ\circ g$, where $g$ is fiber-preserving and orientation-preserving and $ψ\inΨ$, thus reducing the problem to the previously known orientation-preserving case. Through the orientable base-space double covering, we further extend the classification to manifolds with non-orientable base orbifold. |
| title | Fiber-preserving and orientation-reversing involutions of Seifert fibered 3-manifolds |
| topic | Geometric Topology 57S25, 55R05 |
| url | https://arxiv.org/abs/2601.13469 |