Fibre Bundles on Smooth Manifolds

Subjects: Differential Geometry, Topology
Links: Vector Bundles on Smooth Manifolds, Covering Maps

Def: Let $M$ and $F$ be topoplogical spaces. A fiber bundle over $M$ with model fibre $F$ is a topological space $E$ together with a surjective continuous map $\pi :E\to M$ with the property that for each $x\in M$, there exists a neighbourhood $U$ of $x$ in $M$ and a homeomorphism $\mathrm{\Phi }:{\pi }^{-1}[U]\to U\times F$, called a local trivislisation of $E$ over $U$, such that the following diagram commutes

\usepackage{tikz-cd}
\usepackage{amsfonts}
\begin{document}
\begin{tikzcd}[row sep=2cm, column sep=2cm]
\pi^{-1}[U] \arrow[rr, "\Phi"]\arrow[dr, "\pi"']&& U \times F\arrow[dl, "\pi_1"]  \\
& U
\end{tikzcd}
\end{document}

The space $E$ is called the total space of the bundle, $M$ is its base, and $\pi$ is its projection. If $E$, $M$, and $F$ are smooth manifolds with or without boundary, $\pi$ is a smooth map, and the local trivialisations can be chosen to be diffeomorphisms, then it is called a smooth fbre bundle.

A trivial fibre bundle is one that admits a local trivliasiation over the entire base, a global trivialisation. It is said to be smoothly trivial if it is a smooth bundle and the global trivialisation is a diffeomorphism.

Examples:

• Every product space $M\times F$ is a fibre bundle with projection ${\pi }_1:M\times F\to M$, called the product fibre bundle. It has a global trivialisation given by the identity map $M\times F$ to itself, so every product bundle is trivial.
• Every rank-$k$ vector bundle is a fibre bundle with the model fibre ${\mathbb{R}}^k$.
• If $E\to {\mathbb{S}}^1$ is the Möbius bundle, then the image of $\mathbb{R}\times [-1,1]$ under the quotient map $q:{\mathbb{R}}^2\to E$ is a fiber bundle over ${\mathbb{S}}^1$ with model fibre $[-1,1]$.
• Every covering map $\pi :E\to M$ is fibre bundle whose model fibre is discrete.

Properties of Fibre Bundles: Suppose $\pi :E\to M$ is a fibre bundle with fibre $F$.

• $\pi$ is an open quotient map.
• If the bundle is smooth, then $\pi$ is a smooth submersion.
• $\pi$ is a proper map iff $F$ is compact.
• $E$ is compact iff $M$ and $F$ are compact.