Talk:Principal bundle

Transitive action

Resolved

 – Is already mentioned explicitly

Why did you suppress the fact that $G$ acts transitively on the fiber. This is contained implicitly, but should be mentioned explicitly, since it is essential. Hottiger 17:39, 11 April 2006 (UTC)

Measure

For at least some principle bundles, there is a natural measure on he space of connections, is there not? I beleive this is the case when the fibers are compact; less clear of the situation when they're not compact. Would like to see a proper definition. May attempt to do this myself, if/when get around to it. linas 15:51, 20 July 2006 (UTC)

An Algebraic Characterization of Principal Bundles

Both transitivity and freeness of action on the fibres can be directly characterized by defining the quotient on the product bundle, along with the right multiplication by $G$, by the properties p\(pg) = g and p(p\q) = q, where p\q denotes the quotient. This has the advantage of making everything else that follows down the line more transparent. For instance, the connection form simply becomes p\dp, the differential of the quotient by the second argument. In general, the differential of the quotient serves as another equivalent way to define the connection. For a section S, the operator S dS\() + dS S\() is none other than the horizontal lift operator; and the connection relativized to a section is just S\dS. The local decomposition, through a section S, of the principal bundle becomes p = S(pG) S(pG)\p, where pG denotes the projection of p onto the base space. -- Mark, 11 October 2006

It took me a while to understand this, but it seems to be quite clever. Let me expand a bit: the idea is to consider the map from the fiber product $P\times _{M}P$ to M sending a pair (p,q) to the unique element g=p\q of G with pg=q. This is generalizes the difference map (x,y)→y-x from an affine space to the associated vector space of translations. I think that the connection form is here meant to be understood fiberwise, as the generalization of the Maurer-Cartan form. Thus p\dp identifies the vertical bundle of P with the trivial bundle whose fiber is the Lie algebra of G. Does anyone know a reference for this point of view? Geometry guy 16:15, 1 March 2007 (UTC)

Here's an odd suggestion

How about adding an executive summary in the introduction for those of us that are not math students? I realize that such a simplification will not be accurate, yet it remains common practice for other fields to include one, despite it being inaccurate. As this page stands, there's not even a starting point for understanding what this is about if you don't already know it, which is unfortunate when it is linked from non-math subjects. I'm not mathematically impaired, and can pick up the requisite knowledge if you provide a starting point, provided it is possible to recursively apply the process of following the links and absorbing their contents. A "gentle" introduction can ease this process by giving a context and some ideas of how to organize the knowledge. There are many pages that give a lot less context that still get a tag about insufficient context for those not familiar with the field. I haven't watchlisted this, so CCing my talkpage with any replies would be appreciated. Zuiram 04:25, 11 February 2007 (UTC)

This is a fair point and thanks for making it. A lot of mathematics articles suffer from this problem at the moment, because there is so much work to do still. Partly because of this, editors tend to concentrate on "getting the math right" first. For principal bundles, it will be quite a challenge to make an accessible, yet concise, summary, but I hope someone takes it up. Geometry guy 21:14, 21 February 2007 (UTC)

Fibers homeomorphic to G

The page contains: "A principal G-bundle is a fiber bundle π : P → X together with a continuous right action P × G → P by a topological group G such that G preserves the fibers of P and acts freely and transitively on them. This implies that the fiber of the bundle is homeomorphic to the group G itself."

This last statement is not quite true, though it seems to be commonly thrown about. For example, Consider your favorite principle G-bundle, and then replace G by a discrete version of itself. The resulting bundle will still be principle by this definition, but its fibers will still have the topology of the old G, not the discrete topology. If we want the last statement to hold, we had better make it part of the definition (e.g. A principle G-bundle is a fiber bundle with fiber G together with..." --David —Preceding unsigned comment added by 128.95.224.56 (talk) 00:19, 31 December 2010 (UTC)

I agree!!! — Preceding unsigned comment added by 2.236.204.53 (talk) 19:08, 17 March 2013 (UTC)

isnt this already contained in the notion of fiber bundle? Peter Grabs (talk) 18:21, 26 January 2020 (UTC)

I think it would be contained if the article explicitly indicated that principal bundles are fiber bundles with fiber G, but that is left out of the article. Twistar48 (talk) 15:49, 26 March 2024 (UTC)

Examples Needed

This page should discuss more explicit examples/constructions of principal bundles. One nice family of examples are $S^{1}$-bundles from complex line bundles over a space. Since they are always orientable, they are principal bundles. https://mathoverflow.net/questions/144092/is-every-orientable-circle-bundle-principal

These notes discuss the transition functions for $U(1)$-bundles over $S^{2}$ http://ocw.u-tokyo.ac.jp/lecture_files/sci_03/5/notes/en/ooguri05.pdf http://ocw.u-tokyo.ac.jp/lecture_files/sci_03/7/notes/en/ooguri07.pdf

For example, from algebraic geometry, if we have an elliptic curve $E\subset \mathbb {P} ^{2}$, then the line bundles ${\mathcal {O}}(a,b)$ have induced $U(1)$ bundles with interesting chern classes.

Also, this page should discuss theorem 2.5 of riemannian geometry of contact and symplectic manifolds — Preceding unsigned comment added by 71.212.185.82 (talk) 05:02, 24 August 2017 (UTC)

Another good source is https://www.kent.ac.uk/smsas/personal/sk68/ltcc2013/Lecture1.pdf

Lens spaces

• These are principal bundles over projective spaces
• https://math.stackexchange.com/questions/1499631/integral-homology-of-lens-space
• http://www.math.wisc.edu/~maxim/bundle.pdf - good notes

Quantum principal bundles

• https://web.archive.org/web/20200901183245/http://pub.math.leidenuniv.nl/~aricif2/Slides/FrascatiJune2014.pdf + references therein

SU(2) bundles

Look at Atiyah's articles in the twistor page references. He gives criterion for constructing SU(2) bundles over algebraic varieties. The key is to find vector bundles $E\to X$ such that $c_{1}(E)=0$ and $c_{2}(E)$ is the instanton number. This idea comes from looking at SU(2) bundles on $S^{4}$ which is controlled by the transition functions on the equator homotopy equivalent to $S^{3}$, and $SU(2)\cong S^{3}$ so $[S^{3},S^{3}]$ controls the $SU(2)$ bundles on $S^{4}$

Homogeneous Principal Bundles

• https://www3.nd.edu/~snow/Papers/HomogVB.pdf - Useful overview of Root Systems and Characteristic Classes
• https://www.jstor.org/stable/2372795?seq=1 - Homogeneous Vector Bundles (Original paper where techniques in previous link were built)
• https://www.jstor.org/stable/2372747?seq=1 - Part II — Preceding unsigned comment added by 2601:280:8103:EDF0:B1B8:CF07:EE3E:C3B6 (talk) 20:14, 2 January 2020 (UTC)

Overview / intuition correct?

It seems to me that the intuition of a fibre bundle locally being a Cartesian product of a topological space $X$ with a group $G$ is confusing. After all, there is no analog to a local identity element $(x,e)$, no analog to a local group multiplication $(x,g_{1})*(x,g_{2})=(x,g_{1}g_{2})$, and so forth. The way I see it, a better analogy would be a Cartesian product of a topological space $X$ with a set $Y$ with a free and transitive right group action $Y\times G\rightarrow Y$. This naturally generalizes to the group action mentioned later while not introducing unnecessary structure. Thoughts on this? Unfortunately I can't provide a textbook right now explaining it this way. --EduardoW (talk) 09:20, 27 September 2017 (UTC)