# Understanding Fibre Bundles: A Layman's Guide

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• Silviu
In summary, the fiber bundle is an abstract mathematical object that is the union of the direct products of open covers of base space and the fiber. Finding the isotopy group of a given space is not necessary to have a fiber bundle, but is important when applications come into play.
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!

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.

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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

## What is a fibre bundle?

A fibre bundle is a mathematical concept that describes how a space is made up of smaller spaces, each of which looks like a copy of another space. It is used to study different types of spaces and their properties.

## What is the purpose of understanding fibre bundles?

Understanding fibre bundles allows us to better understand the structure and characteristics of different spaces, such as curved surfaces or higher-dimensional spaces. It also has applications in various fields, including physics and engineering.

## How are fibre bundles different from other mathematical concepts?

Fibre bundles are unique in that they can describe spaces that are locally the same but globally different. This means that two spaces may look identical when viewed up close, but have different properties when looked at from a larger perspective.

## Are there different types of fibre bundles?

Yes, there are various types of fibre bundles, including vector bundles, principal bundles, and topological bundles. Each type has its own unique set of properties and applications.

## Do you need advanced mathematics knowledge to understand fibre bundles?

While basic knowledge of mathematics is helpful, it is not necessary to have advanced knowledge to understand fibre bundles. With a bit of patience and effort, anyone can gain a basic understanding of this concept.

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