# principal bundle

Let $E$ be a topological space on which a topological group $G$ acts continuously and freely. The map $\pi:E\to E/G=B$ is called a principal bundle (or principal $G$-bundle) if the projection map $\pi:E\to B$ is a locally trivial bundle.

Any principal bundle with a section $\sigma:B\to E$ is trivial, since the map $\phi:B\times G\to E$ given by $\phi(b,g)=g\cdot\sigma(b)$ is an isomorphism. In particular, any $G$-bundle which is topologically trivial is also isomorphic to $B\times G$ as a $G$-space. Thus any local trivialization of $\pi:E\to B$ as a topological bundle is an equivariant trivialization.

Title principal bundle PrincipalBundle 2013-03-22 13:07:18 2013-03-22 13:07:18 rmilson (146) rmilson (146) 8 rmilson (146) Definition msc 55R10 principal G-bundle