Diffeological vector pseudo-bundles

TL;DR

This paper introduces diffeological vector pseudo-bundles, a generalization of vector bundles that may lack local triviality, and explores their properties, gluing operations, and pseudo-metrics within the diffeological framework.

Contribution

It defines and analyzes diffeological vector pseudo-bundles, including their non-trivial local structure, and develops a gluing method as a substitute for local trivializations.

Findings

Diffeological pseudo-bundles can be non-locally trivial, even when underlying topological bundles are trivial.

A notion of gluing pseudo-bundles is introduced to replace local trivializations.

Interactions between gluing and bundle operations like sum, tensor, and dual are discussed.

Abstract

We consider a diffeological counterpart of the notion of a vector bundle (we call this counterpart a pseudo-bundle, although in the other works it is called differently; among the existing terms there are a "regular vector bundle" of Vincent and "diffeological vector space over X" of Christensen-Wu). The main difference of the diffeological version is that (for reasons stemming from the independent appearance of this concept elsewhere), diffeological vector pseudo-bundles may easily not be locally trivial (and we provide various examples of such, including those where the underlying topological bundle is even trivial). Since this precludes using local trivializations to carry out many typical constructions done with vector bundles (but not the existence of constructions themselves), we consider the notion of diffeological gluing of pseudo-bundles, which, albeit with various limitations that we indicate, provides when applicable a substitute for said local trivializations. We quickly discuss the interactions between the operation of gluing and typical operations on vector bundles (direct sum, tensor product, taking duals) and then consider the notion of a pseudo-metric on a diffeological vector pseudo-bundle.