On extending results of Gluck and Warner on fibrations of spheres by great subspheres

TL;DR

This paper extends Gluck and Warner's 1983 results on fibrations of 3-spheres by great circles to higher dimensions, characterizing when two Hopf fibrations agree on a fiber.

Contribution

It provides a complete characterization of fiber agreement in higher-dimensional Hopf fibrations and discusses obstacles to generalizing Gluck and Warner's results.

Findings

Characterization of when two Hopf fibrations agree on a fiber in higher dimensions

Discussion of barriers to full generalization of the original results

Extension of the bijection between fibrations and distance-decreasing maps to higher spheres

Abstract

In this paper, we build upon the work of Gluck and Warner who showed in 1983 that the set of positively oriented fibrations of a 3-sphere by oriented great circles is in bijection with the set of distance-decreasing maps from the 2-sphere to itself. One approach to generalizing their result to higher-dimensional spheres involves understanding when exactly two Hopf fibrations of $S^{2n-1}$ are guaranteed to agree on a fiber. We give a complete characterization of this phenomenon, and we discuss the barriers which prevent us from obtaining a fully general version of Gluck and Warner's result.