Fiber Bundles: An Overview

Fiber bundles are a central concept in topology and differential geometry, and are crucial in various applications in physics. They provide a way to systematically understand spaces that locally resemble a product of two simpler spaces, but have a more complex global structure.

Definition of a Fiber Bundle

A fiber bundle consists of four main components:

1. Total Space (E): This is the space that the fiber bundle is describing.
2. Base Space (B): A topological space over which the total space is 'bundled'.
3. Projection Map (π): A continuous surjection from $E$ onto $B$. This map 'projects' each point in $E$ down to a point in $B$.
4. Fiber (F): The preimage of a point in $B$ under the projection map $π$, denoted as $π^{-1}(x)$ for $x \in B$, is called the fiber over $x$. Each fiber is homeomorphic to a fixed space $F$.

In mathematical terms, a fiber bundle is denoted as $(E, B, π, F)$.

Local Triviality Property

This requires that for every point in the base space, there exists a neighborhood $U$ such that the preimage of $U$ under the projection map is homeomorphic to the product of $U$ and the typical fiber $F$.

Role of Real and Complex Numbers

In specific types of bundles, particularly vector and principal bundles, real or complex numbers play a crucial role:

• Vector Bundles: These have fibers that are real or complex vector spaces.
• Principal Bundles: In these bundles, the fiber is often a real or complex Lie group.

Quaternionic Fiber Bundles

Fiber bundles can also be defined over other fields like quaternions $\mathbb{H}$:

• Quaternionic Vector Bundles: These have fibers that are quaternionic vector spaces $\mathbb{H}^n$, and appear in specialized contexts in theoretical physics and quaternionic analysis.