Homotopy Type of the Space of Fibrations of the Three-sphere by Simple Closed Curves

TL;DR

This paper determines the homotopy type of the moduli space of smooth fibrations of the three-sphere by simple closed curves, revealing a topological structure akin to either two-spheres or real projective planes depending on orientation.

Contribution

It establishes the homotopy type of the moduli space of fibrations of the three-sphere, relating it to known geometric structures and modeling it via vector fields on the sphere.

Findings

Homotopy type is a disjoint union of two 2-spheres for oriented fibers.

Homotopy type is a pair of real projective planes for unoriented fibers.

The moduli space is a quotient of the diffeomorphism group of the three-sphere.

Abstract

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.