Open quotients of trivial vector bundles

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.

Version 2 fixed a problem with the definition of the closed balls topology (section 5). Version 3 mainly fixed typos and improved presentation

References in corpus (2)

• Etale groupoids and their quantales
• On quantales that classify C*-algebras

Cited by in corpus (3)

• Quantales and Fell bundles
• Linear structures on locales
• On the geometry of physical measurements: topological and algebraic aspects