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Homotopy Type of the Space of Fibrations of the Three-sphere by Simple Closed Curves
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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 Fr\'echet Lie group) by its subgroup of automorphisms of the Hopf fibration, which we show is a smooth Fr\'echet submanifold of the diffeomorphism group. Then we show that the moduli space, already known to be a Fr\'echet manifold by [HKMR12], can be modeled on the concrete Fr\'echet 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.
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Cited by 1 Pith paper
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On Topology of the Infinite-Dimensional Space of Fibrations
The moduli space of smooth fiberings is shown to be a Fréchet manifold, and its homotopy type is computed for circle and torus fiberings on manifolds of dimension at most three.
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