A Non-Trivial $(\mathbf{R},+)$ Principal Bundle over a Contractible Base?

TL;DR

This paper examines a specific quotient space over a contractible base, clarifying its structure as a non-trivial principal pseudo-bundle rather than a strict bundle, due to the failure of local triviality caused by flat functions.

Contribution

It revises previous claims by showing the structure is a principal pseudo-bundle, highlighting the distinction between strict bundles and pseudo-bundles in smooth and diffeological contexts.

Findings

The quotient space is not a strict principal bundle due to flat functions.

The structure has a smooth, free, and fiber-transitive group action.

It exemplifies the boundary between bundles and pseudo-bundles in singular spaces.

Abstract

We investigate the properties of a specific quotient space construction, the "warped projection'" $\pi: W_\alpha \to D_\alpha$, over a smoothly contractible base. In a previous version of this work, it was claimed that this structure constituted a non-trivial principal bundle. We revisit this claim and observe that, due to the existence of flat functions in the smooth category, the projection fails indeed to satisfy the strict condition of local triviality along the plots, required for diffeological bundles. However, the structure remains rich: it possesses a smooth, free, and fiber-transitive group action. Drawing on the concept of vector pseudo-bundles introduced by Christensen and Wu, we propose that this object is best understood as a non-trivial principal pseudo-bundle. This example thus serves to clarify the boundary between strict bundles and generalized pseudo-bundles in the context of singular base spaces.