Vector Bundles over non-Hausdorff Manifolds

TL;DR

This paper extends the theory of real vector bundles to non-Hausdorff manifolds, showing they can be built from standard bundles and classifying line bundles via cech cohomology.

Contribution

It introduces a framework for constructing and classifying vector bundles over non-Hausdorff manifolds using colimits and cohomological methods.

Findings

Vector bundles over non-Hausdorff manifolds can be expressed as colimits of standard bundles.

Formulas relate non-Hausdorff structures to data on Hausdorff submanifolds.

Real non-Hausdorff line bundles are classified by cech cohomology.

Abstract

In this paper we generalise the theory of real vector bundles to a certain class of non-Hausdorff manifolds. In particular, it is shown that every vector bundle fibred over these non-Hausdorff manifolds can be constructed as a colimit of standard vector bundles. We then use this description to introduce various formulas that express non-Hausdorff structures in terms of data defined on certain Hausdorff submanifolds. Finally, we use \v{C}ech cohomology to classify the real non-Hausdorff line bundles.