On the definition of transition maps of a principal bundle

Question

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How are transition maps actting on the trivializations via some continuous left action $G \times F \to F$?

$g_{\alpha \beta}(x)p . \Phi_{\alpha}(x,p) :=\Phi_{\beta}(x,p)$?

Answer 1

What the author means, I believe, is that something just like equation 2.7 holds, except that instead of the transition maps being in $Homeo(F)$, they are in the topological group $G$ (which is a subgroup of $Homeo(F)$. For instance, you can think of line bundles over the circle. If you trivialize over the "north" and "south" "hemispheres" of the circle, then you have to glue near the equator. I'm going to pretend we have to glue at just two points. Perhaps the left gluing map is the identity on the fiber, $\mathbb R$, and the right one is the map $x \mapsto -x$. We call the resulting fiber bundle "the Mobius band". Now each of those gluing maps is a homeomorphism of the fiber, but they're in fact much more special than that: they're each linear. So we've "reduced the structure group to $GL_1(\mathbb R)$." Furthermore, the two maps together form a subgroup homeomorphic to $\mathbb Z / 2 \mathbb Z$, so you could say that we've reduced the structure group to $\mathbb Z / 2 \mathbb Z$.

If we'd parameterized the upper strip differently, the "gluing" map at one end might have been something like $x \mapsto x^3 + x$. By adjusting the parameterization, we might be able to change it to $x \mapsto x$. It's this sort of "simplifying the glue up by changing the parameterization" that's usually called "reducing the structure group."