On transition functions of fiber bundle

In summary, the conversation discusses the construction of a fiber bundle E with g_{\alpha \beta} as its transition function. The speaker explains that the problem lies in the technicalities of the definition and how the bundle is constructed by gluing together trivial bundles over the base B. The projection map pr:E-->B is defined to send [((x,\alpha),y)] to x, making E a fiber bundle. The transition function associated with the trivialisations \Phi_{\alpha} and \Phi_{\beta} is g_{\alpha\beta}, in the sense that \Phi_{\beta}\circ\Phi_{\alpha}^{-1}(x,y)=(x,g_{\alpha\beta}y).
  • #1
kakarotyjn
98
0
I don't understand why the constructed fiber bundle E have g_{\alpha \beta} as its transition function.

The problem is in the pdf file,thank you!
 

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  • #2
It's almost a tautology once you know the little bit of implied information lying behind that statement. Because so far you have defined E. But what about the bundle projection pr:E-->B? Well, it's the natural choice: send [x,y] to x. So now, why is this an F-bundle with structure group G? Well, because pr:E-->B admits local trivialisations over the family {U_\alpha}. Indeed, pr-1(U_\alpha) is homeomorphic to U_\alpha x F via [x,y]-->(x,y). And for U_\beta another guy in that family, what is the transition function [itex]U_{\alpha}\cap U_{\beta} \times F \rightarrow U_{\alpha}\cap U_{\beta} \times F[/itex]? Indeed, it is just (x,y)-->(x,g_{\alpha \beta}y)!
 
  • #3
Thank you quasar,but I'm sorry I still can't understand the connection between the way we construct E and the transition function[tex]g_{\alpha \beta}[/tex].In E, (x,y) is equivalent to (x,g_{\alpha \beta} by definition.But how does this equivalent relation induce that g_{\alpha \beta} is the transition function?Why there is an equivalent relation?

In order to prove g_{\alpha \beta} is the transition function,we need to find fiber-preserving homeomorphisms [tex]\psi_\alpha[/tex] and [tex]\psi_\beta[/tex] for [tex]g_{\alpha \beta}(x)=\psi_\alpha \psi_\beta^{-1}[/tex]

Thank you very much!
 
  • #4
kakarotyjn said:
Thank you quasar,but I'm sorry I still can't understand the connection between the way we construct E and the transition function[tex]g_{\alpha \beta}[/tex].In E, (x,y) is equivalent to (x,g_{\alpha \beta} by definition.But how does this equivalent relation induce that g_{\alpha \beta} is the transition function?Why there is an equivalent relation?

In order to prove g_{\alpha \beta} is the transition function,we need to find fiber-preserving homeomorphisms [tex]\psi_\alpha[/tex] and [tex]\psi_\beta[/tex] for [tex]g_{\alpha \beta}(x)=\psi_\alpha \psi_\beta^{-1}[/tex]

Thank you very much!

Mmh, maybe your problem stems from the technicalities of the definition. Because strictly speaking it is not true that in E, (x,y) is equivalent to [itex](x,g_{\alpha \beta}y)[/itex]. Because E is obtained by quotienting a disjoint union over the index alpha. This means that the elements of E are actually equivalences classes of elements of the form [itex]((x,\alpha),y)[/itex] where [itex]((x',\beta),y')[/itex] is identified to [itex]((x,\alpha),y)[/itex] iff x'=x and [itex]y'=g_{\alpha \beta}y[/itex].

So what we're doing here is we're constructing E by taking a covering {U_\alpha} of the base B and considering the trivial bundles [itex]U_{\alpha}\times F[/itex] over each U_\alpha. Then we glue all of these trivial bundles along fibers over the points where they "intersect" (i.e. over the intersections [itex]U_{\alpha}\cap U_{\beta}[/itex]) by using homeomorphisms coming from the action [itex]G\rightarrow \mathrm{Homeo}(F)[/itex] of the group G on F. This induces a potential "twisting" in the bundle.

For instance, the Mobius bundle can be constructed in this way using G=Z/2Z-->{±IdR} and a covering of S1 of only two open sets {U1,U2}. Then the intersection of U1 and U2 has two connected components. On the first, glue along the fibers following IdR, and on the second, glue along the fibers following -IdR.

With that, define now a projection map pr:E-->B that sends [itex][((x,\alpha),y)][/itex] to x. This is obviously independent of the class so it is well defined. Now E is a fiber bundle because it is trivializable over the U_\alpha's by the map [itex]\Phi_{\alpha}:pr^{-1}(U_{\alpha})\rightarrow U_{\alpha}\times F[/itex] that says "for a class in [itex]pr^{-1}(U_{\alpha})[/itex], pick the representative that belongs to [itex]U_{\alpha}\times F[/itex], say [itex]((x,\alpha),y)[/itex], and send it to (x,y)". Now suppose \beta is another index. How does [itex]\Phi_{\beta}:pr^{-1}(U_{\beta})\rightarrow U_{\beta}\times F[/itex] acts on the same element [itex][((x,\alpha),y)][/itex] of E? Well, it says "pick the representative that belongs to [itex]U_{\beta}\times F[/itex]... well that's [itex]((x,\beta),g_{\alpha\beta}y)[/itex] by definition of the equivalence relation! So send it to [itex](x,g_{\alpha\beta}y)[/itex]."

So you see, the transition function associated with the trivialisations [itex]\Phi_{\alpha}[/itex] and [itex]\Phi_{\beta}[/itex] is [itex]g_{\alpha\beta}[/itex], in the sense that [itex]\Phi_{\beta}\circ\Phi_{\alpha}^{-1}(x,y)=(x,g_{\alpha\beta}y)[/itex].
 
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  • #5
Thank you very much again quasar!Now I really understand it,haha:rofl:
 

1. What is a fiber bundle?

A fiber bundle is a mathematical construct used to describe the topological structure of a space. It consists of a base space, a total space, and a projection map that maps points in the total space to points in the base space.

2. What are transition functions of a fiber bundle?

Transition functions describe how the local coordinate systems of a fiber bundle are related to each other. They are used to understand how the bundle changes as it moves from one point to another in the base space.

3. How are transition functions related to the structure group of a fiber bundle?

The structure group of a fiber bundle is the group of transformations that preserve the bundle's local coordinates. The transition functions are elements of this group, and they determine the structure of the bundle.

4. Can you give an example of a fiber bundle?

One example of a fiber bundle is a Möbius strip. The base space is a circle, the total space is a cylinder, and the projection map maps each point on the cylinder to its corresponding point on the circle. The transition function for this bundle is a half-twist, as the local coordinate systems change when moving from one side of the strip to the other.

5. How are fiber bundles used in physics?

Fiber bundles are used in physics to describe physical fields and their interactions. For example, the electromagnetic field can be described as a fiber bundle with the base space being spacetime, the total space being the set of all possible electromagnetic fields, and the projection map mapping each point in spacetime to its corresponding electromagnetic field. This allows for a deeper understanding of the structure and behavior of physical systems.

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