Deformation and Extension of Fibrations of Spheres by Great Circles

TL;DR

This paper extends the understanding of the topology of the space of great circle fibrations from S^3 to higher-dimensional spheres, showing they deformation retract to Hopf fibrations and that local fibrations extend globally.

Contribution

It generalizes a known result about S^3 to higher dimensions, demonstrating deformation retraction to Hopf fibrations and extension of local fibrations.

Findings

Space of all smooth oriented great circle fibrations deformation retracts to Hopf fibrations.

Every germ of a smooth great circle fibration extends to a global fibration.

The tools used apply to higher-dimensional spheres, broadening previous results.

Abstract

In a 1983 paper with Frank Warner, we proved that the space of all great circle fibrations of the 3-sphere S^3 deformation retracts to the subspace of Hopf fibrations, and so has the homotopy type of a pair of disjoint two-spheres. Since that time, no generalization of this result to higher dimensions has been found, and so we narrow our sights here and show that in an infinitesimal sense explained below, the space of all smooth oriented great circle fibrations of the 2n+1 sphere S^(2n+1) deformation retracts to its subspace of Hopf fibrations. The tools gathered to prove this also serve to show that every germ of a smooth great circle fibration of S^(2n+1) extends to such a fibration of all of S^(2n+1), a result previously known only for S^3 .