I'm having a bit of trouble with principal bundles. Let me be more explicit to explain my problem. I'll start with some definitions.
In the following I'll assume that $G$ is a topological group and that its continuous action over a topological space F is faithful. In this way there is a canonical injection of $G \to S(F)$ where $S(F)$ is the group of $F$ homeomorphisms. For this reason, I'll identify $G$ with a subgroup of $S(F)$
F-structure Let $F$ be a $G$-space and let $X$ be a topological space. An $F$-structure over $X$ is a family $D(F,X)$ of homeomorphisms from $F$ to $X$ such that:
- $\sigma_1^{-1} \circ \sigma_2 \in G \quad\forall \sigma_1, \sigma_2 \in D(F,X)$
- $\sigma \circ g \in D(F,X) \quad \forall \sigma \in D(F,X), \; g \in G$
F-bundle A fiber bundle $\xi = (\pi: E \to B)$ is an $F$-bundle if it assigned a family $D_{F}(\xi)$ of continuous functions from $F$ to $E$ such that:
$\pi(\sigma(v_1))=\pi(\sigma(v_2)) \quad \forall \sigma \in D_{F}(\xi), \; v_1,v_2 \in F$
$D(F,E_b(\xi)):= \{ \sigma \in D_F(\xi) \, | \, \sigma \text{ is an homeomorphism to } E_b(\xi)\}$ is an $F$-structure over $E_b(\xi)$ (where $E_b(\xi)$ is the fiber $\pi^{-1}(b)$) for every $b \in B$.
Principal $G$-bundle In the previous definition, if $F = G$ and $G$ acts on itself through the canonical action $(g,h) \in G \times G \mapsto gh \in G$, we have a principal $G$-bundle.
Principal G-bundle associated to an F-bundle It is the following bundle: $P(\xi) = (\pi': D_F(\xi) \to B)$, definining $\pi'(\sigma) = \pi(\sigma(v_0))$ where $v_0$ is any point in $F$.
(Remark that $\pi$ is the projection of the $F$-bundle $\pi: E \to B$ and that the definition of $\pi'$ makes sense thanks to the $F$-bundle definition where $\pi(\sigma(v))$ doesn't depend on the particular $v \in F$)
Obviously, using the previous definition, $P(\xi)$ fibers are $D(F,E_b(\xi))$.
Let me also remark that $G$ acts over such fibers through the following:
$$(\sigma, g) \in D_F(\xi) \times G \mapsto \sigma \circ g \in D_F(\xi)$$
It is clear that essentially this is something that agrees with the previous definition of principal $G$-bundle, since $G$ is acting on the fibers.
Though, I cannot write this explicitly. I mean, how is $P(\xi)$ a principal $G$-bundle with the given definition? Maybe, I'm missing something obvious, but $G$ should be homeomorphic to the bundle fibers through families $D(G,D(F,E_b(\xi))$ of homeomorphism, but this I cannot see how.
I'd really appreciate it if someone can help me. Thanks :-)