Tags: ? Refs: http://math.mit.edu/~mbehrens/18.906spring10/prin.pdf
principal bundle
Motivation: 
The relationship between fiber bundles and principal bundles: 
Definitions
Definition: A principal \(G{\hbox{-}}\) bundle is a fiber bundle \(F \to E \to B\) for which \(G\) acts freely and transitively on each fiber \(F_b:= \pi^{-1}(b]]\).
I.e. there is a continuous group action \begin{align*}\cdot \in {\mathsf{Top}}{\mathsf{Grp}}(E\times G, E)\end{align*} 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 \(F \to E \xrightarrow{\pi} B\) is universal iff \(E\) is weakly contractible.
Examples
-
Every fiber bundle \(F\to E\to B\) is a principal \(/Aut(F){\hbox{-}}\) fiber bundle.
- In local trivializations, the transition functions are elements of \(G\).
- A covering space \(\widehat{X} \xrightarrow{p} X\) yields a principal \(\pi_1(X){\hbox{-}}\)bundle.
Results
- Every principal \(G{\hbox{-}}\)bundle is a pullback of \({\mathsf{E} G}\to {{\mathbf{B}}G}\).
- A principal bundle is trivial iff it admits a section of a bundle.
- All section of a bundle 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.
-
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 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 \(G{\hbox{-}}\) torsor.
Classification
- If \(G\) is discrete, then a principal \(G\)-bundle over \(X\) with total space \(\tilde X\) is equivalent to a regular covering map with \(\mathop{\mathrm{Aut}}(\tilde X) = G\).
- Under some hypothesis, there exists a classifying space \({\mathbf{B}}G\).