A Non-Trivial $(\mathbf{R},+)$ Principal Bundle over a Contractible Base?
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911277189693440 |
|---|---|
| author | Iglesias-Zemmour, Patrick |
| author_facet | Iglesias-Zemmour, Patrick |
| contents | We investigate the properties of a specific quotient space construction, the "warped projection'" $π: W_α\to D_α$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Non-Trivial $(\mathbf{R},+)$ Principal Bundle over a Contractible Base? Iglesias-Zemmour, Patrick Differential Geometry Primary 58A40, Secondary 55R10, 53C05 We investigate the properties of a specific quotient space construction, the "warped projection'" $π: W_α\to D_α$, 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. |
| title | A Non-Trivial $(\mathbf{R},+)$ Principal Bundle over a Contractible Base? |
| topic | Differential Geometry Primary 58A40, Secondary 55R10, 53C05 |
| url | https://arxiv.org/abs/2508.08708 |