Talk:Fiber bundle

message: Should we make a section on the notion of "locally trivial" bundles? SpiralSource (talk) 10:06, 14 August 2023 (UTC)
thread_title: Local triviality

message: What is the purpose of allowing a generic x in E in the definition, and then talking about a neighbourhood of pi(x)? I'm not confident enough in my understanding to change this immediately, but unless anyone points out something I've been misunderstanding in the next few days, I'd like to change the definition to talk about x in B, and U being a neighbourhood of x, which I think will be clearer. Tatterdemalion1983 (talk) 12:29, 23 November 2019 (UTC)
thread_title: Definition

message: The article should some give some historical context. For example Chern who contributed hugely to the subject, is not mentioned in the article at all, only the Chern class is mentioned once. The first useful construction was in fact the sphere bundle, in the early 40's. Chern was able to give a purely intrinsic proof of Gauss-Bonnet Formula in Integral Formulas for the Characteristic Classes of Sphere Bundles,
Shiing-Shen Chern, Proceedings of the National Academy of Sciences of the United States of America, Vol. 30, No. 9 (Sep. 15, 1944), pp. 269-273,   —Preceding unsigned comment added by Kotika98 (talk • contribs) 03:09, 8 November 2009 (UTC)
thread_title: Chern

message: I find the inconsistency between the article's title and the body (fibre vs. fiber) somewhat annoying. I don't really care whether British or American spelling is used, though considering that there's likely lots of Americanisms in the body, renaming the article to "Fiber bundle" seems better. However, I'm new here, so I'd rather have someone more experienced comment first.
: Rvollmert 23:58, 13 Nov 2003 (UTC)
thread_title:  spelling 

message: Another article to-do: I think there's a statement that isomorphism classes of principle bundles are in 1-1 correspondance to elements of the Čech cohomology group H^1(B,G) where B is the base space and G is the structure group. Or something like that. This follows from the cocycle condition making the bundle a part of an exact sequence. I think the article on covering map tries to say this in the specific case of coverings, but I don't know that it quite hits the mark; its pretty dense as written. I tried to make covering map more accessible, but doubt I suceeded. linas 14:21, 11 May 2005 (UTC)
replies: text: The correct term, by the way, is principal bundles (not "principle" bundles).Daqu (talk) 04:32, 22 January 2015 (UTC)
thread_title: &#268;ech cohomology

message: Could someone confirm that this is the same as a fiber bundle?--MarSch 18:57, 26 November 2005 (UTC)
replies: text: actually it isn't -MarSch 14:36, 25 January 2006 (UTC)
thread_title:  [[fibered manifold]] 

message: * JA: Linas, I get confused by ½-2-alogs on user talk pages and I like to keep issues attached to specific articles, so I will copy your comments here and respond to them after I get some coffee in my veins.  Jon Awbrey 11:32, 6 February 2006 (UTC)
replies: text: Bypassing redirects is not always the right thing to do. In some cases it is unharmful; for example covering map and covering space will probably always point to the same page, so bypassing the redirect does no harm. But in many instances a redirect was created as a substitute for a proper article. For example, section (fiber bundle) currently redirects to fiber bundle, but someone may one day write an actual article on sections of fiber bundles. If someone had previously bypassed the redirect then the link would now point to the wrong article. In general, I try to avoid bypassing redirects altogether. A redirect is harmless and almost unnoticable to the reader. -- Fropuff 17:31, 6 February 2006 (UTC)
replies: text: Hmm, the compute-time saved, if any, by avoiding a redirect is certainly lost several thousand-fold in having to review/edit/watchlist/discuss these thigs ;-) I doubt there's actual CPU-time saving at all: after all, there's some significatnt calculation that has to happen just to parse the wiki markup into something meaningful to begin with, ie. convert it to html. linas 00:11, 7 February 2006 (UTC)
thread_title: Style guidelines, red links, redirects

message: It is well-known that most fiber bundles do not have global sections. The obstructions to building a global section are discussed (or should be discussed, but are not?) in the articles on universal bundle and equivariant cohomology.  In particular, consider the following questions. Suppose I take a local section, and try to extend it, bit by bit, over the whole base space. Eventually, I run into a situation where I can't do this, and a further extension of the section forces it to be "multivalued" over some set of points. Now, for the barage of questions. Lets conduct this by analogy to complex analysis. The square root in complex analysis is a function that covers the complex plane twice, with singularities at 0 and infty, and a cut connecting these two that can be pushed around arbitrarily. It is known that analogous situations occur for fiber bundles: by attempting to promote a local section to a global section, one bumps into singluar points (the so-called Dirac monopole), which are connected by a "cut" (the so-called Dirac string), which can be arbitrarily repositioned. Question: are Dirac monopoles and Dirac string the only kind of singularities that can be encountered? Or are there others? If there are others, do these correspond, in some "clean" fashion, to H_2, H_3, etc?  Is it correct to presume that, if I only knew group cohomology better, then the answers to my questions would be self-obvious? 
:A bundle over a contractible space is always trivial.  If you can find a subspace that differs from your base space by a set of measure zero (can you always do this?  certainly for surfaces you can) then you can have global sections with "branch cuts" or "Dirac strings" in some sense on that leftover set of measure zero.  Oh, by the way, the square root isn't just analogous, it's an example: z --> z2 is a principle Z2 bundle. -lethe talk [ +] 23:51, 20 July 2006 (UTC)
thread_title:  Questions about sections 

message: This article needs a diagram or two in the bundle maps section.  Also, this section needs a discussion of mappings of fibre bundles with structure group. Silly rabbit 10:49, 5 May 2007 (UTC)
thread_title:  Diagram needed 

message: I am a bit concerned with this article since it fails to explain some of the many important aspects of fibre bundles. For a start the article fails to:
* State when two distinct fibre bundles could be isomorphic
* Give more examples of fibre bundles and state some of their properties. For instance, in a fibre bundle with a locally trivial fibration (mapping) p, the preimage of any one point set in the base space is homeomorphic to the fibre F. This is already mentioned but such properties should be given a separate section in the article.
* Instead of naming fibre bundles that are obvious, why not name some bundles which are not covering spaces, but are also quite interesting examples. I can give a few examples but I am not sure how to put them on Wikipedia since I don't know how to put images and write properly in LaTeX. If someone could help with this, I could improve the article in this aspect.
* Also, the article does not really mention the importance of fibre bundles in mathematics (i.e how they are importand and not just why they are important). All the article seems to say currently, is that locally trivial fibrations are just an ordered quadruple of objects that mathematicians play around with. Therefore, the lede of the article needs improving.
* I am not concerned with the level of explanation in the article; this seems fine to me. I think one cannot learn such concepts from someone else; he must rather think of examples and properties of fibre bundles himself. However, if anyone thinks that the explanation in the article is insufficient please respond with a reason.
In fact, if you agree or disagree, could you please reply to this with your opinions. I think that it is time that this article is significantly improved.
Topology Expert (talk) 04:37, 27 August 2008 (UTC)
replies: text: Yah, the article is pretty cheezy.  I changed the examples to be a bit less trivial and I'll poke around at the article some more. IMO one of the main failures is the introduction to the article.  A fibre bundle is not a space, it is a map of spaces.  Rybu (talk) 03:59, 21 September 2008 (UTC)
thread_title:  Problems with the article 

message: For one thing, the projection map needs to be surjective, and typically submersions are not assumed surjective.  Even so, it is still possible to have a submersion in which the fibers are not homeomorphic to each other, let alone not being locally trivial.  There is a theorem of Ehresmann that if π is a surjective submersion whose fibers are connected and compact, then it is a fiber bundle.  But this result is clearly not optimal, and I think it would be very misleading indeed to state it without indicating that it is, in fact a (not so easy) theorem.  siℓℓy rabbit (talk) 11:24, 22 September 2008 (UTC)
replies: text: IMO your comments only indicate that the statement of the theorem wasn't fully-qualified -- ie: they should be qualified, rather than deleted.  Indeed it is an easy theorem -- one that tends to be proven in a 1st course on manifolds, frequently as a homework assignment, or proven as a simple lemma in Milnor's differential topology notes.   One of the problems with this article is that it assumes a rather rigid and not-historical definition of fibre bundle.  A locally-trivial fiber bundle in one of the common historical senses of the word, need not have a fixed homeomorphism type for the fibre.  In particular, the fiber over some points can be empty, provided the base-space is not connected.  Anyhow, I'll see if I can fix up the statement. Rybu (talk) 20:02, 22 September 2008 (UTC)
replies: text: I still find it inadequate.  When I said "not so easy theorem", I meant that it doesn't follow immediately from the definitions of the terms involved that the thing is a fibre bundle.  This makes it particularly poorly suited as an example of a fibre bundle, since there is a not-altogether-trivial theorem needed to give it the structure which it is intended to illustrate.  siℓℓy rabbit (talk) 20:19, 22 September 2008 (UTC)

text: Where does it say that the only examples on Wikipedia can be trivial examples?  And your last edit is wrong.  Surjectivity is implied when the base space is connected and total space compact (unless you're being so pedantic to allow for the empty set to be a manifold).   IMO now you're just making the example overly complicated by quoting the theorem in maximal generality.  Isn't there any effort made towards making the examples easy to state & read? Rybu (talk) 20:43, 22 September 2008 (UTC)
replies: text:  Yes, I suppose it's true that surjectivity is implied by those two conditions.  To be honest, I had a different theorem in mind from the beginning.  Also, it doesn't say anywhere that only trivial examples should be presented.  However, enough detail should be presented along with the example in order to show that it is actually an example of what it is supposed to illustrate.  All I am saying is that I think we should be upfront about this being a theorem.  Otherwise it is really misleading to someone still trying to puzzle out the definition of the thing.  Perhaps this recent addition of yours should be moved to a different section altogether, where such implications can be discussed more fully.  siℓℓy rabbit (talk) 20:55, 22 September 2008 (UTC)

text: I suppose that'd make sense.  These kinds of standards do not seem to be uniformly applied throught the mathematics wikipedia pages -- sometimes examples follow from big theorems, sometimes they're simple and follow from the definition.  Is there a concerted effort to keep examples that follow immediately after definitions to be essentially easy to see?   Rybu (talk) 21:00, 22 September 2008 (UTC)
thread_title:  Not every submersion is a fiber bundle 

message: I don't want to fill the blank in the section heading (I will give you a hint: it starts with 'i') but something should seriously be done about amateurs (with good intentions) writing such "advanced" (compared to other articles) maths articles. The person who wrote most of this article obviously has no idea what a fibre bundle is (but recent edits by more knowledgeable users have improved the article) and nor has any idea as to why it is important. The first sentence is itself rubbish: "a fibre bundle is a space which looks locally like a product space". Nonsense (I don't think I need to explain why). The most appropriate description would be to note that in a sense a fibre bundle is a collection of spaces, all homeomorphic to a particular space F, parametrized by some space B. So in a sense, you are associating to each point of B, a particular space F in a manner that satisfies certain axioms. When there is a free, transitive action by a (topological) group G on the fibre, the bundle becomes a principal G bundle (again there have to be certain conditions met). There is nothing about such bundles in this article. From what I can see, there is absolutely nothing about the long-exact sequence of homotopy group associated to a fibration (another worrying fact) and nothing about Serre fibrations (although I am totally sure whether there should be something on that in this article). I will start with some improvements now but it would be helpful if other people could help out. PST
:Principal bundles are mentioned in the article, both in the current lead and in the section which discussed fiber bundles with a structure group.  As for the first sentence, there is a certain tension between the need to make technical articles accessible and to include meaningful mathematical information in them.  The first sentence is a commonly repeated "introduction" to fiber bundles.  (Actually, I think I wrote that sentence, or an early version of it, borrowing from the manifold article, which is one of the more highly-regarded mathematics articles for its accessibility.)  siℓℓy rabbit (talk) 17:07, 13 January 2009 (UTC)
thread_title:  Did ______ write this article? 

message: better first image to this page? The hairbruse is horripilant :)
--kmath (talk) 03:41, 19 March 2009 (UTC)
replies: text: Of course there are, see further down the article for examples.  But there's not many fibre bundles that embed in 3-dimensional Euclidean space.  If you add the further constraints that you want the fibre bundle not to be a covering space, and if you want it to be non-trivial and a manifold, then pretty much all the examples have been given in the article.  In some sense the "first" non-trivial bundle which is not directly constructed via a covering space construction (such as the mapping cone of a covering space) is the Hopf fibration.  This almost embeds in R^3.   Probably the most pleasant non-manifold examples that are sketchable are things like the bundles over the circle with fibre the cone on a bunch of points.   There's a lot of semi-visualizable highly non-trivial fibre bundles though that we might want to consider.  You can think of RP^3 as this space { (C,p) : C is an oriented great circle on the 2-sphere and p is a point on C }.  So you could think of all the lens spaces L_{2k,1} as being the space { (C, S) : C is an oriented great circle on the 2-sphere and S is a regular k-gon inscribed in the circle }, etc.  These all fibre over the 2-sphere with fibre a circle, but with different monodromy.  Rybu (talk) 04:47, 19 March 2009 (UTC)
replies: text: I think the hairbrush is an excellent lead picture, and I'd like to thank whoever added it, and say I'm glad it was kept.  I'm a graduate electrical engineer, so I'm not totally unexposed to higher math, but its hard to visualize from abstract definitions in exactly what ways a "fiber bundle" is analogous to a fiber in physical experience.  The hairbrush picture and its caption really nailed it for me, and I think it would be even more important for  nontechnical readers.  Pictures of the more interesting examples you mention could be added, but I think the hairbrush should be kept.   Thanks again. --ChetvornoTALK 02:47, 31 March 2018 (UTC)
thread_title: are there...

message: The article contains the following sentence:
:"A fiber bundle (E, B, π, F) is often denoted  to indicate a short exact sequence of spaces."
I am aware that $F\to E \stackrel{\pi }{\to }B$ is a common way to notate fibre bundles. I think, however, that it's just meant to indicate that this is somewhat like a short exact sequence. I don't think that it really is a short exact sequence, as I don't see a canonic morphism $F\to E$. I could imagine that there are cases where $B\times F \to E \to B$ is a short exact sequence. Shouldn't the article be clear about the point, whether this either _is_ a short exact sequence, or whether it's not? -- 131.220.192.246 (talk) 13:58, 16 June 2009 (UTC)
replies: text: IMO that comment is misleading.  A fibre bundle is a fibre bundle.  There is no notion of short exact sequence of spaces.   The analogy is the homotopy long exact sequence of a fibration.  No point being vague and misleading. Rybu (talk) 08:46, 17 June 2009 (UTC)
replies: text: I think so too. But I fear that I just don't get it and that the problem lies with me. I think so mainly, because I find this notion in the literature. Just now I saw it in Lawson, Michelsohn: Spin Geometry. -- 78.54.224.191 (talk) 12:08, 17 June 2009 (UTC)

text: When you say you just saw it, do you mean Lawson has an explicit defininition of a short exact sequence of spaces?  Rybu (talk) 04:22, 19 June 2009 (UTC)
replies: text:  No, I guess he's using it notationwise, but maybe I'm wrong. In his context, the $P$s are principal bundles, X is the manifold, E a vector bundle above X. See Spin Geometry, 1st edition from 1989 [there is a second edition from 1994], p79: "...Then from the fibration $O_n \to P_O(E) \to X$, there is a exact sequence. 0\to H^0(..)\to H^0...." and he's also using it in commutative diagrams as on page 80: Z_2 \to Spin _n \to SO_n and below that Z_2(same as above) \to P_{Spin _n}(E) \to P_{SO}(E) and below that the manifold X. Namely: There are arrows in the commutative diagram from $SO_n$ to $P_{SO}(E)$ and from $Spin _n$ to $P_{Spin _n}(E)$ -- 78.48.213.58 (talk) 10:27, 19 June 2009 (UTC)

text: ah, okay.  IMO it looks like he's just being lazy.  It's a very common thing for mathematicians to do.  When there's such a strong analogy between objects in two different subject areas, and when they're so closely linked together you start to blur the notions. For a wikipedia article I strongly encourage avoidance of this kind of laziness.  Because if you start calling fibre bundles short exact sequences of spaces you might as well start calling covering spaces galois extensions of spaces and then you have overlapping analogies that aren't entirely coherent. Rybu (talk) 17:19, 19 June 2009 (UTC)
replies: text:  Is this a common issue in mathematics? I wouldn't call it lazy to use an analogy at all - it's often more work finding a more accessible concept with threads to connect where you need them - only lazy to omit that it is an analogy, where the threads may unravel, etc. I guess I'm not sure if the intended audience is someone brushing up or learning from scratch, but without some kind of analogy I'm usually clicking through some links more than once before both the individual terms and their relationship to one another really stick with me. I mean... analogies can't be that detrimental if the standing thumbnail is a hairbrush, right? (For what it's worth, my name for 'fiber bundle' is 'hedgehog particle' - you're welcome.) I was swayed somewhat by Dijkstra's "On the Cruelty of Really Teaching Computing", but personally I've found that analogies pave the way to understanding without them. Sobeita (talk) 20:21, 31 March 2018 (UTC)
thread_title:  F->B->E just notation or real math? 

message: The section titled Möbius strip begins as follows:
"Perhaps the simplest example of a nontrivial bundle E is the Möbius strip."
It would be hard to come up with a nontrivIal bundle simpler than the fibre bundle given by the circle double-covering the circle.  This may be defined by the map f: S1 → S1 via f(z) = z2, where S1 is the unit circle in the complex plane ℂ.  The fibre here is the simplest nontrivial group, the group of two elements, and this bundle is identical with its associated principal bundle, which is also the associated principal bundle of the Möbius strip.
This bundle is also only 1-dimensional with total space a manifold, whereas the Möbius strip is 2-dimensional with total space a manifold-with-boundary. It should definitely be presented here as the simplest bundle.
It is also very misleading to characterize a fibre bundle as being defined just by its total space. There are many examples to show that the same space can be the total space of various inequivalent nontrivial fibre bundles.  For example, for any positive integer n, the circle is the total space of the fibre bundle ξn defined by the bundle projection  pn: S1 → S1 via pn(z) = zn, and all of these are inequivalent. (Of course, there is also the group of the bundle to consider, but it is not necessary to be that technical at this point.) 
Therefore, I propose that the statement quoted be changed to read:
"One of the simplest fibre bundles has total space the Möbius strip, with bundle projection equal to the projection of the strip onto its core circle, and with fibre equal to the closed unit interval."
Daqu (talk) 00:59, 17 January 2015 (UTC)
thread_title:  Simplest fibre bundle?  I don't think so

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thread_title:  External links modified 

message: In the section on the Möbius strip, it reads: "locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space)".  It seems to me that the word "cut" here is ambiguous, and what may be clearer would be "creating a boundary on the surface".  If one were really to cut the Möbius strip, that would be creating two separate spaces -- each side of the strip.  Making a single cut in the cylinder would *not* yield two equivalent spaces.  But setting up a boundary on (one side of) the Möbius strip would be equivalent to setting up a boundary on the cylinder.
Zeroparallax (talk) 20:34, 26 September 2018 (UTC)
thread_title:  "cutting" the Mobius strip 

message: I think the example of the Möbius strip would be easier to follow with a labeled diagram, rather than identifying the relevant features using textual descriptions. Can someone please check whether these figures, created in Mathematica based on the existing Möbius strip diagram, are correct?
 Stevvers (talk) 20:28, 11 May 2024 (UTC)
thread_title:  Möbius figures