I'm having a hard time understanding principal bundles.
- I understand that a fiber bundle can be defined $(E, B, \pi, F)$ where $E, B, F$ are topological spaces and $\pi: E\to B$ is a surjection. For a fiber bundle, for each $b\in B$ there is a neighborhood $U\supset b$ such that $\pi^{-1}(U)\subset E \simeq U\times F$ via homomorphism $\phi$. Also $\text{proj}_1\circ \phi = \pi$.
- I've seen multiple definitions for the principal bundle. One definition is that a principal bundle is a fiber bundle $(E, B, \pi, G)$ where $G$ is a topological group. But in addition, it seems to be important that there is a free and transitive group action of $G$ on $E$ which preserves fibers of $B$ within $E$.
I have a few questions.
- Is the definition I've given for a principal bundle above good? I've seen many different definitions. Some of them involve quotients/orbit spaces, some do not indicate that a principal bundle is defined to have typical fiber $G$, some involve a structure group, etc. I would appreciate if answers to this question include the definition of a principal bundle preferred by the author, possibly with a reference for that definition.
- THE MAIN QUESTION: Why is it necessary to specify a group action of $G$ on $E$? Why couldn't we just define a principal bundle to be a fiber bundle whose typical fiber is a group $G$?
Disclaimer: I'm still learning, this is my stab at an answer, but correction/improvements are much appreciated.
I think the key here is that the goal with a principal bundle is to attach a $G$-torsor to each point of the base space, not to attach the group $G$ to each point of the base space. There is a bit of confusion, however, because a $G$-torsor is always isomorphic to $G$, so you can't attach a $G$-torsor without kind of attaching $G$ as well. This is what makes the definitions a bit confusing.
But then the question arises, why not define a principal bundle as a fiber bundle $(E, B, \pi, T)$ where the fiber $T$ is a $G$-torsor? I think the answer here is that it is not enough for the group $G$ to act independently on/within the various fibers $E_b\subset E$ for $b\in B$, but rather, it is necessary that the group acts continuously on the total space, i.e. globally and continuously across fibers as well.
For this, we require a continuous group action of $G$ on $E$ which preserves fibers and which acts freely and transitively on fibers. One possible definition which does this is:
In fact, I think this definition suffices despite not providing ANY information about the fibers $F$ other than the fact that is a fiber of some fiber bundle. It could indeed be proven as a simple theorem that $F$ is homeomorphic to a $G$-torsor. But then, of course, we could show $F$ is also homeomorphic to $G$.
So this means there is freedom in the definition to (1) specify nothing about $F$, (2) specify that $F$ is a $G$-torsor, or (3) specify that $F$ is $G$. I think this freedom in the definition is where some of my confusion comes from. For example, I think of a vector bundle as attaching a vector space to each point in the base space. For my intuition, should I think of a principal bundle as attaching a group or a torsor to each point of the base space? I think the answer is that it's probably best to think of a principal bundle as attaching a $G$-torsor to each point in the base space. So for this I think the definition that makes no claim about the fiber $F$ is best. It can then be left as a theorem that $F$ is homeomorphic to a $G$-torsor and to $G$ itself.