Definition of structure group of a fiber bundle

Question

In Spanier AT, A structure group $G$ of a fiber bundle is defined as follows:

Let $(E,B,F,p)$ be a fiber bundle. Given a space $F'$ define a collection $\Phi=\{\varphi\}$ of homeomorphism $\varphi:F\to F'$. A fiber bundle is said to have structure group $G$ if each fiber $p^{-1}(b)$ has a $G$ structure $\Phi(b)$ such that there exists an open covering $\{U\}$ of $B$ and for each $U\in \{U\}$, a homeomorphism $\varphi_U:U\times F\to p^{-1}(U)$ such that for $b\in U$, the map $F\to p^{-1}(b)$ by $x\mapsto \varphi_U(b,x)$ is in $\Phi(b)$.

1. What is $\Phi(b)$? for $\varphi\in\{\varphi\}$, $\varphi(b)$ is not defined.
2. It says '... there exists an open covering $\{U\}$...'. Is this open covering possibly different from the open covering in the definition of fiber bundle?
3. What is the meaning of 'a $G$ structure $\Phi(b)$'?

I'm very confusing right now. Can somebody help?

Answer 1

In this paragraph, after first defining "$G$-structure" in general, and then after choosing $G$ to be a particular group of homeomorphisms of $F$, the paragraph continues by defining what it means for a fiber bundle $(E,B,F,p)$ to have "structure group $G$". That definition starts with

... each fiber $p^{-1}(b)$ has a $G$ structure $\Phi(b)$...

To reword this, as $b$ varies over the points of the base space $B$, we are given $G$-structures, one on each fiber $p^{-1}(b)$. The given $G$-structure on the fiber $p^{-1}(b)$ is denoted $\Phi(b)$. Thus $\Phi(b)$ is a collection of homeomorphisms from $F$ to $p^{-1}(b)$ which satisfy conditions (a) and (b).

Regarding your question 2, the definition of a fiber bundles requires the existence of a "fiber bundle atlas", which is my term for an open cover $\{U\}$ of $B$ and homeomorphisms $\phi_U : F \times U \to p^{-1}(U)$ satisfying some compatibility conditions. There is a common feature of atlases in different contexts (i.e. of fiber bundle atlases; of atlases of differentiable structures; and of many other types of atlases):

• atlases are not unique;
• you can compare two atlases and examine whether they are compatible with each other (which means, roughly speaking, that their union is also an atlas);
• when you are given one atlas, you can usually use it to produce a lot of other compatible atlases.

So the literal answer to your question 2 is "Yes:" Assuming that you started with a particular fiber bundle atlas, the open cover in the definition of "structure group $G$" might well be part of a different fiber bundle atlas. However it will always be compatible with the given atlas you started with.