Understanding Fibre Bundles: A Layman's Guide

[Silviu]

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!

 

[fresh_42]

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.

 

[WWGD]

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.

 

[fresh_42]

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.

 

[WWGD]

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.

 

[fresh_42]

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.

 

[WWGD]

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.

 

[WWGD]

Sorry for my confusion, Fresh_Meister