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Let $\pi: P \rightarrow M$ be a principal $G$-bundle whose fibers are a Lie group $G$. In Hamilton's Mathematical Gauge Theory he states that

The group $G$ is called the structure group of the principal bundle.

I was left thinking that the term structure group is another way to refer to the fiber of a principal bundle.

However, I have also come across other resources (such as this question: What is the structure group of the tangent bundle?) that gives a more general definition where the structure group can be completely different than the fiber. From what I have understood of their definition, let $\pi: E \rightarrow M$ be a fiber bundle with fiber $F$. If $u$ is in the domain of two trivializations $\phi_1: E \rightarrow M \times F$ and $\phi_2: E \rightarrow M \times F$, with images $\phi_1(u) = (x, f_1)$ and $\phi_2(u) = (x, f_2)$, then the structure group $G$ of the fiber bundle is a group whose group action relates these two images, namely $f_1 = g \cdot f_2$ for some $g \in G$.

I am now a little confused on what the structure group is exactly and how it relates to the fiber of a fiber bundle, especially in the context of principal bundles. In the case where we have a principal $G$-bundle is the structure group always the same as the fiber (and hence the Lie group) $G$ or can it be generalized to another group?

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  • $\begingroup$ Are you familiar with associated bundles of a principal bundle? For instance: the tangent bundle $TM$ of a smooth manifold $M$ has as fibers the tangent spaces $T_pM$. It is an associated bundle to the so-called frame bundle $FM$ of $M$, which is a principal $\mathrm{GL}(\dim(M),\mathbb R)$-bundle. $\endgroup$ Commented Jul 17, 2023 at 9:28
  • $\begingroup$ @NicoZimmer I am familiar with frame bundles but I have not yet gotten to the section on associated bundles in the book. $\endgroup$
    – CBBAM
    Commented Jul 17, 2023 at 9:30
  • $\begingroup$ My point being, that the fibers of a principal $G$-bundle are always diffeomorphic to the Lie group $G$. However the fibers of a generic fiber bundle (for instance an associated bundle) typically don't have to be. $\endgroup$ Commented Jul 17, 2023 at 9:34
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    $\begingroup$ I'd be careful with the phrasing. "The" structure group isn't unique as adding additional structure to a fiber bundle can be equivalent to reducing "the" structure group to a subgroup. However in the setting of principal $G$-bundles we already provide the relevant group to begin with and any associated bundle to this principal $G$-bundle will have the very same Lie group $G$ as structure group. Generally I'd recommend to think about the structure group as additional structure and not a property of a fiber bundle per se. $\endgroup$ Commented Jul 17, 2023 at 9:54
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    $\begingroup$ Regarding your last question: Of course there is a relation between "the" structure group and the fiber $F$, since the structure group usually (if the group acts faithfully from the left on the fiber $F$) can be taken to be the diffeomorphisms of $F$ or a subgroup thereof (assuming we work in the smooth category of course). But other than that the fibers do not have to be diffeomorphic to the structure group etc. $\endgroup$ Commented Jul 17, 2023 at 10:06

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What’s in the link is actually not a more general definition, it is more restrictive. Here’s what Hamilton writes; immediately after giving the definition of a principal bundle, he says

Remark 4.1.3: The classic references [133] (Steenrod) and [81] (Husemoller) use the term fibre bundle in a more restrictive sense; see Remark 4.1.15.

and remark 4.1.15 says the following:

Remark 4.1.15: Some references, such as [133] and [81], use the term fibre bundle more restrictively. If the topological definition in these books is transferred to a smooth setting, the definition amounts to assuming that the transition functions of a bundle atlas are smooth maps to a Lie group $G$, acting smoothly as a transformation group on the fibre $F$, instead of maps to the full diffeomorphism group $\text{Diff}(F)$ of the fibre: \begin{align} \phi_{ji}:U_i\cap U_j&\to G\\ x&\mapsto \phi_{jx}\circ\phi_{ix}^{-1}. \end{align} Equivalently, a fibre bundle is with this definition always an associated bundle in the sense of Remark 4.7.8.

So, fiber bundle as defined by Hamilton (and also other authors) does not come equipped with a group, whereas in the other references of Steenrod and Husemoller (let’s just take Husemoller for example) the definition is as the associated fiber bundle:

Definition.[Husemoller, Definition of Fiber Bundle]

Let $\xi=(X,p,B)$ be a principal $G$-bundle and let $F$ be a left $G$-space. The relation $(x,y)s=(xs,s^{-1}y)$ defines a right $G$-space structure on $X\times F$. Let $X_F$ denote the quotient space $(X\times F)/G$ and $p_F$ the factorization of $X\times F\to X\to B$ by the projection $X\times F\to X_F$. With this notation, the bundle $(X_F,p_F,B)$, denoted $\xi[F]$, is called the fiber bundle over $B$ with fiber $F$ (viewed as a $G$-space) and associated principal bundle $\xi$. The group $G$ is called the structure group of the fiber bundle $\xi[F]$.

Reiterating, note that although in this source one calls $G$ “the structure group of $\xi[F]$”, the very definition of fiber bundle is restrictive, in the sense that a fiber bundle is defined to be an associated bundle of a principal bundle (whereas Hamilton requires it to be a triple $(X,p,B)$ where $p$ is surjective and smooth/continuous and satisfy a local-triviality condition… other names for this object include locally-trivial (smooth/continuous) bundle).

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  • $\begingroup$ I think it's really a matter of semantics at this point. I'd still argue that $\mathrm{Diff}(F)$ should be named "the" structure group of the fiber bundle, even with Hamilton's more general definition. $\endgroup$ Commented Jul 17, 2023 at 11:36
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    $\begingroup$ @NicoZimmer I agree, I was just pointing out where the semantic differences were (actually pointing out where Hamilton already points them out). $\endgroup$
    – peek-a-boo
    Commented Jul 17, 2023 at 11:37
  • $\begingroup$ although, I personally don’t actively think of the diffeomorphism group as being the structure group in the sense that, because it’s infinite-dimensional I’d rather steer clear of having to interpret a statement like “the transition maps $U_i\cap U_j\to \text{Diff}(F)$ are smooth” in a literal sense by equipping the diffeomorphism group with a smooth structure etc etc. but yea semantics at this point $\endgroup$
    – peek-a-boo
    Commented Jul 17, 2023 at 11:44
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    $\begingroup$ @CBBAM sure, but really its semantics; just keep reading and things will fall into place. WHat you really need to know are the main concepts (local triviality, bundle charts, trivial vs non-trivial, key examples etc) $\endgroup$
    – peek-a-boo
    Commented Jul 17, 2023 at 18:38
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    $\begingroup$ @Callum The general linear group does not always make sense as a structure group. If we're talking about vector bundles or frame bundles, however, sure. $\endgroup$ Commented Jul 17, 2023 at 20:39

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