Talk:Circle bundle

Principal
Seems to me like we should move this page to principal circle bundle. Shouldn't circle bundle just refer to a fiber bundle where the fibers are circles (a special case of a sphere bundle). In that case, not every circle bundle is principal. For example the Klein bottle can be viewed as a (nontrivial) circle bundle over the circle but the structure group is only Z/2Z. -- Fropuff 03:26, 10 October 2006 (UTC)


 * Do you know if each circle bundle is principal apart from that example? --kiddo 00:16, 29 October 2006 (UTC)

No, one can construct numerous examples using the associated bundle construction. Let G be a topological group which has a effective action on the circle that is not equivalent to left translations of the circle. Any dihedral group for example. Given a principal G-bundle over any space X one can then construct an associated circle bundle over X which is not a principal circle bundle. The Klein bottle is the simplest example. Here G = Z/2 and the associated principal bundle is the nontrivial double cover of the circle by itself. -- Fropuff 04:41, 31 October 2006 (UTC)

Error in classification
This article is somewhat incorrect. Circle bundles are not classified by the Euler class. Circle bundles are classified by the homotopy-class of the map $$M \to BO_2$$. Since there is the extension $$SO_2 \to O_2 \to \mathbb Z_2$$, if the composite $$M \to BO_2 \to B\mathbb Z_2$$ is null, *then* the map is classified by the associated map to $$BSO_2 = BU_1$$. Basically the article skips the possibility that the fibers may not have a coherent orientation -- for example the unit tangent bundle of the projective plane is a circle bundle that the article ignores. Rybu (talk) 01:49, 6 May 2014 (UTC)

Relationship to electrodynamics
It is stated that "the Aharonov–Bohm effect is not a quantum-mechanical effect (contrary to popular belief), as no quantization is involved or required in the construction of the fiber bundles or connections." It is my understanding that wave-particle duality (i.e. that the electron is described by a wave-function, and can exhibit wave-interference) is inherent in the analysis of this effect, wave-particle duality being quintessentially quantum. If there really is a way to show how the AB effect is purely classical, it would benefit this article greatly to make references to the appropriate literature. I want a bear as a pet (talk) 17:59, 21 February 2023 (UTC)


 * P.S. as far as I know though, the AB effect is a quantum effect, and I think it should be stated as such. I want a bear as a pet (talk) 18:01, 21 February 2023 (UTC)

Not-necessarily-principal circle bundles
The section Classification contains this statment:

"Thus, for the more general case, where the circle bundle over M might not be orientable, the isomorphism classes are in one-to-one correspondence with the homotopy classes of maps $$M \to BO_2$$."

Currently the article does not state anything about what kind of space $$BO_2$$ is, and nothing about how to compute the homotopy classes of maps $$M \to BO_2$$. Nor does it contain any examples of a computation of equivalence classes of not-necessarily-principal circle bundles over some interesting topological spaces (like compact surfaces). Nor does it contain any links to such information.

Furthermore: The classification of arbitrary circle bundles should be given significantly more emphasis in this article than information about the relation of circle bundles to electrodynamics or to Deligne complexes.

There is, of course, nothing wrong with including information about electrodynamics or Deligne complexes.

But the subject of how to classify arbitrary circle bundles is much more fundamentally relevant to this article and needs to be addressed here. And earlier in the article than the other two subjects.

I hope that someone knowledgeable about this subject can add (a lot) more detail to this statement about classifying not-necessarily-principal circle bundles, i.e., arbitrary circle bundles.