Understanding Fibre Bundles: A Layman's Guide

  • Context: Undergrad 
  • Thread starter Thread starter Silviu
  • Start date Start date
  • Tags Tags
    Bundles Fibre
Click For Summary

Discussion Overview

The discussion revolves around the concept of fibre bundles, focusing on their definition, visualization, and specific examples, particularly in the context of the circle and the fibre [-1,1]. Participants seek to clarify the relationship between the total space, base space, and projection function, as well as the role of gluing maps in the structure of fibre bundles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about whether the fibre bundle refers to the projection function, total space, or something else, and questions the terminology used.
  • Another participant asserts that a fibre bundle encompasses the total space, base space, projection map, and fibres, emphasizing that the relationship is local rather than global.
  • A participant introduces the concept of gluing maps, suggesting that they are necessary for closing the circle in the example of fibres around the base space.
  • Some participants argue that gluing is not essential for defining a fibre bundle, suggesting that the definition can be understood without geometric concepts.
  • There is a discussion about the implications of isotopy and homotopy in the context of fibre bundles, with differing views on their necessity in the definition.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of gluing maps and geometric concepts in defining fibre bundles. There is no consensus on whether these elements are essential or merely applicable in specific contexts.

Contextual Notes

Some participants note that the understanding of fibre bundles may depend on specific applications, and there are unresolved questions regarding the role of isotopy and homotopy in the definition.

Silviu
Messages
612
Reaction score
11
Hello! I am having some troubles understanding fibre bundles and I would be really grateful if someone can explain them to me in layman terms (at least how to visualize them). To begin with, I am not sure what is the fibre bundle, is it the projection function, or the total space (or something else)? I found an example of calculating the fibre bundles of ##S^1## with the fibre ##F= [-1,1]## and the results says that the fibre bundles are a cylinder or the Mobius strip. Based on this, I would say that the total space and the fibre bundle are the same thing, but then why 2 names for the same mathematical object? Also, as far as I understood, the fibre bundle is the union of the direct products of open covers of base space and the fibre, but this seem to straightforward for the definition fibre bundles have. So can someone explain to me how should I think of them? Thank you!
 
Physics news on Phys.org
What you ask is a bit long to type in as well as it can be found on Wikipedia or many other sources which you can google. My attempt to answer it has been: https://www.physicsforums.com/insights/pantheon-derivatives-part-iii/
So the short answer is: A fiber bundle is all of them (total space E, base space X, projection map π, fibers F). The crucial part in its understanding is, that it is in general not globally E = F x X but only locally. To call it union is even less exact.
 
fresh_42 said:
What you ask is a bit long to type in as well as it can be found on Wikipedia or many other sources which you can google. My attempt to answer it has been: https://www.physicsforums.com/insights/pantheon-derivatives-part-iii/
So the short answer is: A fiber bundle is all of them (total space E, base space X, projection map π, fibers F). The crucial part in its understanding is, that it is in general not globally E = F x X but only locally. To call it union is even less exact.
I think to complete the bundle the remaining instructions are the gluing maps up to isotopy ( maybe homotopy, not sure). You have a line " floating" about each point in the circle, by definition. At some point, in order to close the circle, two lines will have to come together " glued " to each other in certain ways. The "certain ways" is a choice of map between two lines , up to isotopy/homotopy. So, for each isotopy class of maps you have a choice of bundle up to bundle morphism. One class of self-maps is given by the identity, which gives you the cylinder: do the same parametrization in each line, say f(t)=t; 0<t<1 and map each t to itself, or the map t--> 1-t , which gives you the Mobius . I think finding the isotopy group of the Real line should do it here: either order-preserving or order-reversing are the two classes.
 
Last edited:
As far as I see it, this is not necessary to have a fiber bundle. It's important when applications come into play. Therefore sections are used which regain the topology in a way.
 
fresh_42 said:
As far as I see it, this is not necessary to have a fiber bundle. .
Sorry, I don't know what you mean by this. I was addressing his example of fiber bundle fiber ##[-1,1]## over ##\mathbb S^1 ##.. Sorry for quoting you instead.
 
WWGD said:
Sorry, I don't know what you mean by this.
A fiber bundle is two topological spaces, a continuous projection with fibers as its preimages. I don't see where gluing should come into play rather than by special cases for applications. The entire topic is certainly inspired by tangent bundles and atlases. But for the pure definition, I don't think geometric concepts (lines, homotopies etc.) are needed.
I was addressing his example of fiber bundle fiber ##[-1,1]## over ##\mathbb S^1 ##.. Sorry for quoting you instead.
Uh yes, sure.
 
fresh_42 said:
A fiber bundle is two topological spaces, a continuous projection with fibers as its preimages. I don't see where gluing should come into play rather than by special cases for applications. The entire topic is certainly inspired by tangent bundles and atlases. But for the pure definition, I don't think geometric concepts (lines, homotopies etc.) are needed.

.

I was referring to the specific case of the OP : the fibers circle around the base and must ultimately be glued together at some point.
 
Sorry for my confusion, Fresh_Meister
 

Similar threads

Graduate Is Fibre Bundles Cartan's Generalization of Klein's Erlagen Program?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Product Space vs Fiber Bundle: Understanding the Difference
  • · Replies 15 ·
Replies
15
Views
5K
Graduate Understanding the Fibre Bundle Structure of Mobius Band
  • · Replies 11 ·
Replies
11
Views
5K
Graduate Fiber Bundle Basics: An Overview of Penrose's Twisted Bundles
  • · Replies 35 ·
2
Replies
35
Views
11K
Graduate On local trivializations and transition functions of fibre bundles
  • · Replies 5 ·
Replies
5
Views
7K
Graduate Elementary questions about fibre bundles
  • · Replies 5 ·
Replies
5
Views
5K
Graduate What is a fiber bundle and how is it related to smooth manifolds?
  • · Replies 16 ·
Replies
16
Views
5K
Graduate On the relationship between Chern number and zeros of a section
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad About the maximal extension of local charts on a manifold
  • · Replies 8 ·
Replies
8
Views
2K
Graduate Tangent Bundle: Why It's Important
  • · Replies 2 ·
Replies
2
Views
3K