2
$\begingroup$

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? .

$\endgroup$
5
  • 1
    $\begingroup$ 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. $\endgroup$
    – Lee Mosher
    Jul 11 at 14:42
  • 1
    $\begingroup$ Regarding question 3, did you look up Spanier's definition of $G$-structure? $\endgroup$
    – Lee Mosher
    Jul 11 at 14:42
  • $\begingroup$ @LeeMosher I just edit the post. The definition of $G$-structure is not defined separately. The word '$G$-structure' first used in that definition. $\endgroup$
    – love_sodam
    Jul 11 at 15:24
  • $\begingroup$ Actually, by my reading "$G$-structure" is defined separately, right there in that same paragraph, previous to the definition of "structure group $G$". $\endgroup$
    – Lee Mosher
    Jul 11 at 15:37
  • $\begingroup$ @LeeMosher Oh, I see. But I still don't know what $\Phi(b)$ is. $\endgroup$
    – love_sodam
    Jul 11 at 15:43
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.