Open quotients of trivial vector bundles

TL;DR

This paper introduces a general framework for classifying quotient vector bundles over topological spaces using hyperspaces of linear subspaces, extending classical classifications of locally trivial bundles.

Contribution

It generalizes the classification of vector bundles by using hyperspaces with natural topologies, encompassing Banach bundles and bundles with continuous norms.

Findings

Hyperspaces classify various quotient vector bundles.

Bundles of constant finite rank are locally trivial.

The framework includes Banach bundles as a special case.

Abstract

Given an arbitrary topological complex vector space $A$, a quotient vector bundle for $A$ is a quotient of a trivial vector bundle $\pi_2:A\times X\to X$ by a fiberwise linear continuous open surjection. We show that this notion subsumes that of a Banach bundle over a locally compact Hausdorff space $X$. Hyperspaces consisting of linear subspaces of $A$, topologized with natural topologies that include the lower Vietoris topology and the Fell topology, provide classifying spaces for various classes of quotient vector bundles, in a way that generalizes the classification of locally trivial vector bundles by Grassmannians. If $A$ is normed, a finer hyperspace topology is introduced that classifies bundles with continuous norm, including Banach bundles, and such that bundles of constant finite rank must be locally trivial.