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

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.