Abstract Definition of Bundle

Let $G$ be a Topological group such that $G\times F\to F$ is a continuous action.

Definition of Fiber Bundle with structure group and fiber $F$

A fiber bundle with structure group $G$ and fiber $F$, denote as $\left(G,F\right)$-bundle is a bundle with following structures. It has a map $\pi :E\to B$ from total space to base. There is an atlas $\mathcal{A}=\left\{\phi :U\times F\to {\pi }^{-1}\left(U_{\phi }\right)\right\}$. Also, there is a Transition functions $\theta =\left\{{\theta }_{\phi ,\psi }:U_{\phi }\cap U_{\psi }\to G|\phi ,\psi \in \mathcal{A}\right\}$ such that

• $\left\{U_{\phi }\right\}$ is open cover of $B$.
• the diagram

commutes.

• $\psi (b,f)=\phi (b,{\theta }_{\phi ,\psi }(b)f)$.
• the cocyle condition ${\theta }_{\phi ,\psi }(b)={\theta }_{\phi ,\chi }(b){\theta }_{\chi ,\psi }(b)\in G$.

Examples

	$G$	$F$	$\xi$	Comment
1.	1	Any $F$	Trivial Bundle	Effective
2.	$G{L}_n({\mathbb{R}}^n)$	${\mathbb{R}}^n$	$n$-plane vector bundle
3.	$O(n)$	${\mathbb{R}}^n$	Vector bundle with metric
4.	$G{L}_n^{+}({\mathbb{R}}^n)$	${\mathbb{R}}^n$	Oriented Vector bundle
5.	$Sym(F)$	Discrete	Covering Space
6.	Discrete $G$	$G$	Regular, $G$-cover
7.	$Spin(n)$	${\mathbb{R}}^n$	Vector bundle with spin structure	$Spin(n)$ is not effective.
8.	$O(n)$	$O(n)$	Frame bundle over $M$	In Riemannian manifold, with fiber ${\pi }^{-1}(b):=$ Orthonormal basis of Tangent vectors.