Fibre Bundles over Contractible spaces

[ajb]

Does any one know who to prove that fibre bundles over contractible spaces are trivial?

 

One method woud be to use the Homoptopy axion for bundles. Do you know how easy that is to prove?

 

I was wondering if there is a more geometric method of proving it. It seems obvious that on say R^n we could use the global coordinates of R^n as a local trivialisation. Thus only one trivialisation would be needed and hence we have a trivial bundle. Can this be proved?

 

Any ideas and references welcome.

 

Cheers

AJB:confused:

[matt grime]

I think this question goes (far) beyond the average knowledge of the user base of this forum. Try posting it at http://www.physicsforums.com instead where there are more likely to be people who can answer it. My knowledge of fibre bundles is sketchy at best. I don't particularly see why arguing aobut bundles over R^n says anything bout bundles over other spaces which are not R^n, for instance. Presumably, the argument is to show that any bundle of degree m over your base B is homotopic to the space {pt}xR^m, and then invoke this theorem you stated to deduce the final result. Not knowing what the homotopy axiom for bundles is I couldn't say with any confidence this is what you're supposed to do. I imagine mathwonk at physicsforums know the best argument.

[Cap'n Refsmmat]

I love your high confidence in us, matt

[matt grime]

I'd prefer for you to call it realism. The level of mathematics questions asked (and answered) here is relatively low. If we need a scale, let's use the US one:normally the hardest questions are 'easy' freshman or sophmore calc. questions. This question is more 'grad school qualifier' standard. I.e. probably the question has come up to a good final year undergrad student or later.

 

I think it only reasonable to point the OP at a better (i.e. wider) source of knowledge than we have here. I'd be interested to know how many users here can eve define what a fibre bundle is, for instance. From the answers I've observed here in the past I would suggest at most 3 others, all of whom also post at physicsforums, maybe one of whom actually knows the answer.

 

I've never really studied fubre bundles, and though the question, or rather what the question is getting at, seems sort of clear, I can't provide an answer other than to say he/she should ask somewhere else.

[ajb]

I was thinking about R^n as I had manifolds as the base space in mind. Once I assume the homoptopy it is easy to prove. Basically, it allows you to identify a bundle over a contractable space with a bundle over a point, which is trivial.

 

I will look at the website you suggest Matt.

 

Cheers

[Chuck Edge]

I'm no mathematician, but as I understand it given spaces X and Y and a surjective map [imath]f[/imath] from X to Y, a fiber is an inverse map [imath]f^{-1}[/imath] back to X. If X is contractible, then it is homotopy equivalent to a point, in which case the bundle space has only one element and is therefore trivial. Am I on the right track?

[matt grime]

There is slightly more to a fibre bundle than that (such as transition maps and a structure group). The bundle space does not have only one element (not that I'm that sure exactly what you mean by bundle space: X is the total space, Y the base space in your notation). The base space is homotopy equivalent to a one point set, obviously. The point is to show that, for contractible spaces, homotopy equivalence is the same as homeomorphism or whatever.

[Chuck Edge]

There is slightly more to a fibre bundle than that (such as transition maps and a structure group). The bundle space does not have only one element (not that I'm that sure exactly what you mean by bundle space: X is the total space, Y the base space in your notation). The base space is homotopy equivalent to a one point set, obviously. The point is to show that, for contractible spaces, homotopy equivalence is the same as homeomorphism or whatever.

 

That should fiber space, not bundle space, right? And since [imath]f[/imath] from X to Y is surjective, a fiber bundle is the set of inverses back from Y to X, right? The way I see it, if space Y is a point or contracted to a point and [imath]f[/imath] is surjective, then there's only one fiber back to X. This gives us a product space [imath]f \circ f^-1[/imath] that's trivial. I have no idea where I'm going with this.

[Martin]

There is slightly more to a fibre bundle than that (such as transition maps and a structure group). The bundle space does not have only one element (not that I'm that sure exactly what you mean by bundle space: X is the total space, Y the base space in your notation). The base space is homotopy equivalent to a one point set, obviously. The point is to show that, for contractible spaces, homotopy equivalence is the same as homeomorphism or whatever.

 

brave soul, to be teaching fibre bundles in this context.

a brave soul deserves all the assistance one can readily supply!

 

proof of desired theorem in a textbook by CalTech guy Jerry Marsden

Chapter 3, supplement 3.4B

http://www.cds.caltech.edu/oldweb/courses/2002-2003/cds202/textbook/MTA_Ch3.pdf

around page 44-45 of the PDF chapter here

It is nominally on page 168-169 of the book

the theorem is 3.4.35

 

Let pi:E ->B be any C⁰ fibre bundle over a contractible space B. then E is trivial.

 

proof is short, only 4 or 5 lines, but uses a lemma. however the idea of the proof is nice and simple. one studies a pullback fibre bundle over the base B x [0,1] defined by the contraction homotopy. The pullback restricted to Bx{0} is trivial and the lemma says that this percolates out to the whole pullback bundle. So the whole pullback bundle is trivial and therefore the part over Bx{1} is trivial. which is the original thing. qed.

 

Marsden says that the theorem extends to C^k. he just proves it for continuous but it's true for smooth.

 

marsden chapter 3 is generally useful, has many definitions illustrations and exercises. is a real textbook. enjoy y'all

[ajb]

Thanks Martin, I will have a look at the PDF you suggest.

 

I know how to prove it using the homoptopy axiom, it is quite simple really. I have no idea how you would prove the axiom.

 

AJB

[Martin]

Thanks Martin' date=' I will have a look at the PDF you suggest.

 

I know how to prove it using the homoptopy axiom, it is quite simple really. I have no idea how you would prove the axiom.

 

AJB[/quote']

 

BTW I'm not 100 percent sure that chapter is by Marsden, it doesnt have his name on it. But I remember from several years back I was reading some chapters by Marsden that looked just like that----a book on diff. geom, diff topol. and dynamical systems----maybe a draft of the book he wrote with Ralph Abraham (Manifolds Tensor Analysis and Applications).

 

I havent read this thread in detail, just happened in and saw a couple of posts. I kind of suspect that Marsden's proof does not use the "homotopy axiom" explicitly----I think it is a very elementary nuts-bolts beginning-textbook kind of proof. I'm curious: how about you STATE the "homotopy axiom"?

 

I dont want to get in the way here. I dont usually visit math forum. But when you say "fibre bundle" it gets my interest. I am interested in quantum gravity, especially recent stuff of Laurent Freidel. What are you interested in? Your profile says PhD student in UK and I can't make out if all the abbrevations mean anything.

[ajb]

The title of my PhD topic is "geometric constructions in quantum field theory".

 

I was told to go away and read up on fibre bundles and gauge theory, which I have done now, up to a few proofs etc. So thats why I want to be VERY clear on trivial bundles etc.

 

We are also interested in the BRST symmetries, these are a clever way of dealing with gauge theories by promoting the gauge parameter to be a dynamical anticommuting field. It is particulary useful when dealing with gauge fixed actions and quantum gauge theory. There is a lot of (super)geometry behind this which is of interest.

[Martin]

PhD topic is "geometric constructions in quantum field theory".

Then I will show you a few Freidel abstracts, to put Laurent Freidel on the map for you if you dont know of him already

 

 

1. gr-qc/0607014 [abs, ps, pdf, other] :

Title: Particles as Wilson lines of gravitational field

Authors: L. Freidel, J. Kowalski--Glikman, A. Starodubtsev

Comments: 19 pages

 

3. hep-th/0604184 [abs, ps, pdf, other] :

Title: Towards a solution of pure Yang-Mills theory in 3+1 dimensions

Authors: Laurent Freidel, Robert G. Leigh, Djordje Minic

Comments: 12 pages

 

 

5. gr-qc/0604016 [abs, ps, pdf, other] :

Title: Hidden Quantum Gravity in 3d Feynman diagrams

Authors: Aristide Baratin, Laurent Freidel

Comments: 35 pages, 4 figures

 

 

7. hep-th/0512113 [abs, ps, pdf, other] :

Title: 3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory

Authors: Laurent Freidel, Etera R. Livine

Comments: 4 pages, to appear in Phys. Rev. Letters, Proceedings of the conference "Quantum Theory and Symmetries 4" 2005 (Varna, Bulgaria), v2: some clarifications on the Feynman propagator and slight change in title

 

12. hep-th/0501191 [abs, ps, pdf, other] :

Title: Quantum gravity in terms of topological observables

Authors: Laurent Freidel, Artem Starodubtsev

Comments: 19 pages

 

Freidel will be in Cambridge around September 4-8 at a Templeton-funded workshop where there will also be a lot of Big-Name Old Guys (hawking. gerard 't hooft, alain connes, shahn majid) and younger string celebs too.

but he is not string and not a celebrity or even very well known outside quantum gravity. he DOES relate the feynman diagrams of Quantum Field Theory to the spacetime geometry of quantum gravity, though

 

so he is in there close to your PhD topic------relation of QFT to geometry.

[ajb]

I have come across Freidel, but i have never actually read any of his papers.

 

I take it you are talking about the noncommutative geometry workshop at the Newton Institute. I won't be going, but i do know some of the participants.

 

Also as a side note, my supervisor did his PhD under A. Schwarz, who is one of the invited speakers.

[Martin]

I have come across Freidel' date=' but i have never actually read any of his papers.

 

I take it you are talking about the noncommutative geometry workshop at the Newton Institute. I won't be going, but i do know some of the participants.

 

Also as a side not, my supervisor did his PhD under A. Schwarz, who is one of the invited speakers.[/quote']

 

I am not so interested in NCG per se. Freidel has co-authored one paper with Shahn Majid (earlier this year) but he is not into NCG either.

I wouldnt necessarily go to that NCG workshop myself, but mentioned it just in case you were.

 

Freidel has a version of spinfoam quantum gravity that gives Feynman diagrams in the zero gravity limit, that is in the limit as G -> 0.

 

non-string QG is a very small field, only on the order of 100 researchers worldwide, so everything is miniaturized, but in those terms, Freidel's work is having a huge effect now.

 

the essential thing is, I think, that non-string QG as a field is beginning to push for a connection with QFT and the standard model.

the stereotype is that the field was only interested in quantizing spacetime dynamics (with some kind of non-descript generic matter) so it was just quantizing General Relativity. Now they want to do more. They want to find GEOMETRICAL REASONS why the different numbers of and types of particle.

Aims in a way are more similar to string theory except no extra dimensions and also they seem to be making more noticeable progress at the moment.

 

what kind of research does your advisor do? (given that he was a student of Albert Schwarz)

 

Schwarz was doing topological field theory papers 5 years ago. this could be exciting. Schwarz does some string but is far from doing ALL string.

 

What several QG people are working on is basically perturbing around a topological field theory called BF THEORY. (one nickname for it is "beef") they work with constrained or extended BF. Freidel but also others.

E.g. some recent papers by John Baez and collaborators.

[ajb]

Majid does quantum groups and related stuff if I remember rightly.

 

I know about loop quantum gravity, but it is not my research topic. My understanding (not being an expert) is that including matter was an issue.

 

My supervisors are Hovik Khudaverdyan and Ted Voronov. I would call their work supermathematics and supergeometry. Recient work of interest has been on odd-symplectic geometry and odd Laplacians. These things are all connected with the antifield formulism.

 

Topological field theory was something that we might look at also. Not sure exactly what, but probabily something to do with exactly solvable models and gravity in 3 and 2 dimensions.

 

I expect my research will be something related to this.

[Martin]

 

My supervisors are Hovik Khudaverdyan and Ted Voronov.

 

I looked at Voronov's stuff on arxiv. Moscow and then UC Berkeley in 1996 or so. I saw Quantum Groups, and category theory.

reminds me of John Baez interests.

 

In any case you seem to have found yourself an interesting line of research for your PhD thesis! good luck with it! (I will be away from this thread for a few days---also do not want to get in matt's way since he's in charge)

Ciao:-)

[matt grime]

I'm not in charge. It's good to have a discussion on this level in here for a change. Keep it up. My knowledge of string theory is slightly better than my knowledge of elementary properties of fibre bundles, although I am more of al algebraist than anything, so 1-d TFTs are the same as A-infinity algebras. I tend to come in to this subject as someone who understands (as much as is possible) triangulated categories (the subject of my PhD) and is curious as to what it is the string theorists are doing with them, or trying to do with them (D-, A-, and B-branes).