# Definition of structure group of a fiber bundle

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? • Is your copy of that definition letter-by-letter exactly? I see grammatical errors which I doubt occured in Spanier, and those errors make it hard to determine the best way to answer your question. Jul 11 at 14:42
• Regarding question 3, did you look up Spanier's definition of $G$-structure? Jul 11 at 14:42
• @LeeMosher I just edit the post. The definition of $G$-structure is not defined separately. The word '$G$-structure' first used in that definition. Jul 11 at 15:24
• Actually, by my reading "$G$-structure" is defined separately, right there in that same paragraph, previous to the definition of "structure group $G$". Jul 11 at 15:37
• @LeeMosher Oh, I see. But I still don't know what $\Phi(b)$ is. Jul 11 at 15:43

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.