#fibre bundle

Hopf Fibration [Heinz Hopf, 1931 | Mathematics Discovery] ~Gif

In much the same way that a manifold is an object that looks locally like \(\mathbb{R}^n \), a fibre bundle is an object that looks locally like the product of two manifolds. The best example to have in mind is a Möbius strip.

Crocheted by Sophie Nuttall.

The Möbius strip, \( M \), can be thought of a circle (aka \( S^1 \)) onto which a line segment, \(I\), is attached at each point. We call the circle the base space and the whole Möbius strip the total space. One way to describe which points in \( M \) lie `over' a given point on the circle (i.e. in the line segment attached to that point) is to define a function, \( \pi: M \rightarrow S^1 \), that takes a point in \( M \) to the point on \( S^1 \) that it lies over. The line segment over a point \( p \in S^1\) is then given by \(\pi^{-1}\{p\}\). This is known as the fibre over \(p\). Looking at some open set, \(U\), in the circle, we can see that the part of the Möbius strip that lies `above' \(U\) i.e. \(\pi^{-1}(U)\) is topologically the same as (homeomorphic to) \( U \times I \). In symbols: \( \pi^{-1}(U) \cong U \times I\).

The best drawing of a Möbius strip that I could muster. The base space is the black circle going along the middle. The fibres are the red line segments attached to the base circle.

The definition of a fibre bundle goes something like this:

Definition (Fibre Bundle)

A fibre bundle is a quadruple \((E,B,\pi,F)\).

The objects \(E, B, F\) are topological spaces: \(E\) is known as the total space; \(B\) is known as the base space and \(F\) is known as the fibre for the bundle.

The object \(\pi\) is a continuous and surjective map \(\pi:E \rightarrow B \) known as the projection map.

We require that, for each \(x \in B\), there is an open set, \(U \subset B\), containing \(x\), such that \(\pi^{-1}(U)\) is homeomorphic to \(U \times F\).

Let \(\varphi\) be this homeomorphism. For \( p \in E \), denote \(\varphi(p) = ( q, r )\) so that \(q \in U\) and \( r \in F\). We also demand that \( \pi(p) = q \).

The last bullet point basically says that the point in \(U\) `under' \(p \in E\) is also the left-hand entry of \(\varphi(p)\). This ensures that the homeomorphism is keeping the correct points `over' each other and isn't mixing them around.

Geotagged photos (e.g. flickr) and text (e.g. twitter) associate data to a particular point on the globe × time. In other words, a fibre bundle over S²×T.