Homotopy Type of the Space of Fibrations of the Three-sphere by Simple Closed Curves
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911088042311680 |
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| author | Deturck, Dennis Fang, Ziqi Gluck, Herman Lichtenfelz, Leandro Merling, Mona Wang, Yi Yang, Jingye |
| author_facet | Deturck, Dennis Fang, Ziqi Gluck, Herman Lichtenfelz, Leandro Merling, Mona Wang, Yi Yang, Jingye |
| contents | We show that the moduli space of all smooth fibrations of a three-sphere by simple closed curves has the homotopy type of a disjoint union of a pair of two-spheres if the fibers are oriented, and of a pair of real projective planes if unoriented, the same as for its finite-dimensional subspace of Hopf fibrations by parallel great circles. This moduli space is the quotient of the diffeomorphism group of the three-sphere (a Fréchet Lie group) by its subgroup of automorphisms of the Hopf fibration, which we show is a smooth Fréchet submanifold of the diffeomorphism group. Then we show that the moduli space, already known to be a Fréchet manifold by [HKMR12], can be modeled on the concrete Fréchet space of vector fields on the three-sphere which are "horizontal" and "balanced" with respect to a given Hopf fibration, and see how the structure of this moduli space helps us to determine its homotopy type. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_01185 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homotopy Type of the Space of Fibrations of the Three-sphere by Simple Closed Curves Deturck, Dennis Fang, Ziqi Gluck, Herman Lichtenfelz, Leandro Merling, Mona Wang, Yi Yang, Jingye Geometric Topology Algebraic Topology Differential Geometry 58B05 (Primary) 58B10, 57R30 (Secondary) We show that the moduli space of all smooth fibrations of a three-sphere by simple closed curves has the homotopy type of a disjoint union of a pair of two-spheres if the fibers are oriented, and of a pair of real projective planes if unoriented, the same as for its finite-dimensional subspace of Hopf fibrations by parallel great circles. This moduli space is the quotient of the diffeomorphism group of the three-sphere (a Fréchet Lie group) by its subgroup of automorphisms of the Hopf fibration, which we show is a smooth Fréchet submanifold of the diffeomorphism group. Then we show that the moduli space, already known to be a Fréchet manifold by [HKMR12], can be modeled on the concrete Fréchet space of vector fields on the three-sphere which are "horizontal" and "balanced" with respect to a given Hopf fibration, and see how the structure of this moduli space helps us to determine its homotopy type. |
| title | Homotopy Type of the Space of Fibrations of the Three-sphere by Simple Closed Curves |
| topic | Geometric Topology Algebraic Topology Differential Geometry 58B05 (Primary) 58B10, 57R30 (Secondary) |
| url | https://arxiv.org/abs/2508.01185 |