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I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following:

A bundle is a tuple $(E,B,\pi)$ where $E,B$ are topological spaces and $\pi: E\to B$ is a continuous surjection.

A fiber bundle is a tuple $(E,B,\pi,F)$ where $(E,B,\pi)$ is a bundle and $F$ is a topological space such that there's an open cover $\{U_\alpha\}$ of $B$ together with homeomorphisms $\varphi_\alpha : \pi^{-1}(U_\alpha)\to U_\alpha\times F$ with the property that if $\pi_1 : U_\alpha\times F\to U_\alpha$ is the projection onto the first factor, then $\pi_1\circ\varphi_\alpha = \pi$.

That is perfectly fine, but then one notices the following: if $\varphi_\alpha(p) = (\pi(p), \tilde{\varphi}_\alpha(p))$ then $\tilde{\varphi}_\alpha : \pi^{-1}(U_\alpha)\to F$. Then for each $p \in U_\alpha$ we can consider $\tilde{\varphi}_{\alpha, p} = \tilde{\varphi}_\alpha |\pi^{-1}(p)$ and show that this mapping is an homeomorphism from $\pi^{-1}(p)$ onto $F$.

If $U_\alpha\cap U_\beta$ is then non-empty, we have $\varphi_\alpha$ and $\varphi_\beta$, and hence for each $p\in U_\alpha\cap U_\beta$ two homeomorphism $\tilde{\varphi}_{\alpha, p}, \tilde{\varphi}_{\beta, p} : \pi^{-1}(p)\to F$ and thus we can form the mapping

$$g_{\alpha\beta}(p) = \tilde{\varphi}_{\alpha, p}\circ \tilde{\varphi}_{\beta, p}$$

which is a homeomorphism of $F$. That is all fine, but the book I'm reading says these maps are really important, in the sense that they give the structure of the fiber bundle, like the twisting of Möebius band. That's the first problem: how can I see these maps really give the structure of the fiber bundle? I really can't realize the importance of those changes of trivialization.

Second, the book introduces in a little confusing way the "Structure Group". At first it seems that when we want to show something is a bundle, the Structure Group must come with it. Like, $TM$ the fiber bundle of $M$ comes with $GL(n,\mathbb{R})$, but then the book talks in a way that it seems that the Structure Group is really determined by those transition functions only. The book even says: "if the structure group has one element the fiber bundle is trivial". That's fine, but then it doesn't make sense anymore to pick something like $GL(n,\mathbb{R})$ since the group will come from the trivializations themselves.

So my questions really are:

  1. What is the real importance of the transition functions? How to understand them intuitively?

  2. What really is the structure group? What it is really meant to represent and where does it really comes from?

Thanks very much in advance.

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1 Answer 1

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The first thing to do is to compute some of the transition maps. A point where the Mobius band twists has a transition map that's multiplication by a negative element of $\mathbb{R}$. To see this, just write down a trivialization via two charts.

This leads to the fact that a bundle is trivial, i.e. homeomorphic to the Cartesian product of the base and the fibre, if and only if its structure group "can be reduced to the trivial group." If you haven't encountered bundle maps yet just think of the scare-quoted phrase as reflecting the fact that the transition maps for a nontrivial bundle like the Mobius band can't possibly all be the identity, or even all be positive.

It's indeed a bit of a foggy matter whether the structure group should be given as part of the structure of the bundle or whether it's something computed from the structure. In the latter approach one says the structure group of a vector bundle like $TM$ has to be contained in $GL(V)$, but needn't equal it; in the former approach the transition functions take values in the structure group but needn't take on every element of that group. It's not conceptually an important distinction.

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