The fiber bundle is defined as
A fiber bundle is defined as the tuple $(E, B, \pi)$ where $\pi: E \to B$ is a continuous surjective map from topological space $E$ to topological space $B$. Furthermore, there is a topological space $F$, called the fiber, such that for each $b\in B$ there is an open neighborhood $U\subset B$ with $b\in U$ such that $\pi^{-1}(U)\subset E$ is homeomorphic to $B\times F$ via homeomorphism $\varphi: \pi^{-1}(U)\to B\times F$ and $\text{proj}_1\circ \varphi = \pi$.
$\pi^{-1}(\{b\}) \subset E$ is called the fiber of $b$.
One definition of a principal bundle is
The tuple $(E, B, \pi)$ is a $G$-principal bundle if it is a fiber bundle and if there is a continuous action $\lhd:E\times G \to E$ which preserves the fibers in $E$ (that is $x\in \pi^{-1}(b)$ then $x \lhd g \in \pi^{-1}(b)$ for $g\in G$) and which acts freely and transitively on the fibers in $E$.
But another definition or property I've seen for principal bundles over smooth manifolds is (e.g. in the Schuller lectures)
The tuple $(E, B, \pi)$ is a $G$-principal bundle if there is a smooth free action $\lhd: E\times G \to G$ and $\pi:E\to B$ is isomorphic as a smooth bundle to $\rho: E\to E/G$ where $\rho$ is the normal equivalence relation projection map where $\rho(e)$ for $e\in E$ maps $e$ to $[e]$, the equivalence class of $e$.
Since $\pi:E\to B$ is a smooth bundle then $E$ and $B$ are manifolds, $\pi$ is a smooth map, and $\varphi:\pi^{-1}(U)\to B\times F$ is a diffeomorphism.
Wikipedia also says that similar properties fully characterize the principal bundle.
Here are my questions.
- Is there any reason the second principal bundle definition can't be relaxed to topological fiber bundles instead of smooth fiber bundles?
- If the second principal bundle definition can and is weakened to be for topological fiber bundles then are the two definitions equivalent?
- If the first principal bundle definition is strengthened to apply to smooth manifolds then are the two definitions equivalent?
