A singular Serre-Swan theorem via tepui fibrations

TL;DR

This paper extends the classical Serre-Swan theorem to include singular vector bundles modeled by tepui fibrations, enabling the realization of all finitely generated modules over smooth functions, including singular cases.

Contribution

It introduces tepui fibrations as a new geometric framework for singular vector bundles, broadening the scope of the Serre-Swan theorem beyond projective modules.

Findings

Defined singular vector bundles using tepui fibrations

Modeled singular foliations and groupoids with tepui fibrations

Extended the Serre-Swan correspondence to non-projective modules

Abstract

The classical Serre-Swan theorem asserts that any finitely generated projective module over the algebra $C^\infty(M)$ of smooth functions of a manifold $M$ can be realized as the sections of a vector bundle over $M$. In this article, we extend this theorem beyond the projective case by introducing a notion of singular vector bundle whose sections can realize all finitely generated $C^\infty(M)$-modules, up to invisible elements. We introduce tepui fibrations as the underlying geometric objects of these singular vector bundles, and show how these tepui fibrations can model singular foliations, their holonomy groupoids, and singular subalgebroids.