---
date: 2022-01-15 21:49
modification date: Saturday 15th January 2022 21:49:49
title: principal bundle
aliases: [principal]

---

Tags: 
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Refs:
<http://math.mit.edu/~mbehrens/18.906spring10/prin.pdf>


# principal bundle

![](attachments/Pasted%20image%2020220221004330.png)

Motivation:
![](attachments/Pasted%20image%2020220213020813.png)

The relationship between [fiber bundles](fiber%20bundle) and principal bundles:
![](attachments/Pasted%20image%2020220213021001.png)

# Definitions

![](attachments/Pasted%20image%2020210613123515.png)

**Definition:** 
A **principal $G\dash$ bundle** is a [fiber bundle](fiber%20bundle) $F \to E \to B$ for which $G$ acts [freely](freely) and [transitively](transitively) on each fiber $F_b:= \pi^{-1}(b]]$. 

I.e. there is a continuous group action
$$\cdot \in \Top\Grp(E\cross G, E)$$ 
such that for every $f \in F_b$ and $g\in G$,
 
- $g\cdot f \in F_b$ 
- $g\cdot f \neq f$.

A principal $G$ bundle is a locally trivial free $G$-space with orbit space $B$. 

**Definition:** A principal bundle $F \to E \mapsvia{\pi} B$ is **universal** iff $E$ is [weakly contractible](weakly%20contractible).

![](attachments/Pasted%20image%2020210510012449.png)
![](attachments/Pasted%20image%2020210613123624.png)
![](attachments/Pasted%20image%2020210613123710.png)

# Examples
- Every [fiber bundle](fiber%20bundle) $F\to E\to B$ is a principal $/Aut(F)\dash$ fiber bundle. 
	- In local trivializations, the transition functions are elements of $G$.
- A [covering space](covering%20space.md) $\hat X \mapsvia{p} X$ yields a principal $\pi_1(X)\dash$bundle.

# Results
- Every principal $G\dash$bundle is a pullback of $\EG \to \BG$.
- A principal bundle is trivial iff it admits a [section of a bundle](section%20of%20a%20bundle). 
- All [section of a bundle](section%20of%20a%20bundle) always exists.
- Each $F_b \cong G \in \text{TopGrp}$, which may also be taken as the definition.
- Each $F_b$ is a [homogeneous space](homogeneous%20space.md).
- Although each fiber $F_b \cong G$, there is no preferred identity element in $F_b$. 
	- Locally, one can form a [section of a bundle](section%20of%20a%20bundle) by choosing some $e\in F_b$ to serve as the identity, but the fibers can only be given a global group structure iff the bundle is trivial. 
	- So each fiber $F_b$ is a $G\dash$ [torsor](torsor.md).

## Classification

- If $G$ is discrete, then a principal $G$-bundle over $X$ with total space $\tilde X$ is equivalent to a [regular](regular) [covering](covering.md) map with $\Aut(\tilde X) = G$. 
- Under some hypothesis, there exists a [classifying space](classifying%20space.md) $\B G$.