Definition of principal $G$-bundle might be missing details or have implicit assumptions on actions on each trivializing open set

Question

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An Introduction to Manifolds by Loring W. Tu (Volume 1).

I refer to Section 27.1 (part 1), Section 27.1 (part 2) and Section 27.1 (part 3).

Firstly:

1. I believe the book has no explicit definition for an action $\mu$ to be "transitive" and neither does Volume 1. I think this is okay for the book since Proposition 27.6 is not (explicitly) used later on in the book.

   • 1.1. If this wouldn't be okay for the book, then I would ask how, if possible, we could deduce from Tu's definition of principal $G$-bundle that from the action $\mu: P \times G \to P$, we get that $\mu(P_x \times G) \subseteq P_x$, where $P_x := \pi^{-1}(x)$, which is saying something like $\mu$ is fiber-preserving, such that we can define an action $\mu_x: P_x \times G \to P_x$ and then begin to discuss whether or not each $\mu_x$ is transitive.

   • 1.2 Even though I'm not asking (1.1), what I'm about to ask has a similar underlying problem.

2. Anyway, I assume the definition that an action $\mu$ is "transitive" is the one here, assume that definition is equivalent to the one on Wikipedia and assume that that both definitions are equivalent to "for each $x \in M$, the map $\mu_x : G \to M, \mu_x(g) = \mu(x,g)$, is surjective, where $\mu: M \times G \to M$ is the right action of $G$ on $M$".

Now:

3. Tu's definition of principal $G$-bundle doesn't say anything about transitive or fiber-preserving, but it may be equivalent to a definition with transitivity (see here). I mean that transitive or fiber-preserving could be somehow deduced from Tu's definition (as stated). Tu's definition is possibly the "Definition 3" in the previous link). I guess the alternative is that Tu made a mistake in the definition of principal $G$-bundle.

4. I actually notice that for each $U \in \mathfrak U$, while we are given an explicit action $\sigma_U: U \times G \times G \to U \times G$, which is $\sigma_U((x,h),g)=(x,hg)$, we are not given an explicit definition of the action $\zeta_U: P_U \times G \to P_U$, where $P_U := \pi^{-1}(U)$.

   • 4.1. Edit: Oh wait that was kind of wrong. What I meant was to say that $\zeta_U$ is not even declared to exist in the first place. I really think the text is unclear here. I think the text should've said something like "$G$ acts on $U \times G$ (in the way of $\sigma_U$), and then $G$ acts on $\pi^{-1}(U)$ in such a way that $\phi_U$ is invariant". Otherwise, it seems kinda weird that you just say a map is equivariant even though you haven't declared the existence of an action on both domain and range. It just seems that somehow the action $\mu$ on $P$ induces $\zeta_U$'s.

     • 4.1.1. Edit: Probably, there should even be some prior proposition that starts with "given a map $f: N \to M$ and action $\zeta$ by $G$ on $N$ we can define an action $\sigma$ by $G$ on $M$" or that starts with "given a map $f: N \to M$ and action $\sigma$ by $G$ on $M$ we can define an action $\zeta$ by $G$ on $N$" and then the next part would be "that makes $f$ equivariant" and then there might be another proposition or some exercise that says that the defined $\zeta$ or $\sigma$ is unique. I'm thinking of something similar to the pullback metric, from earlier in the book.

     • 4.1.2. Edit: A comment of autodavid: In the definition of principal $G$-bundle, the way in which $G$ is acting on $P$ should make the trivialization maps $G$-equivariant when restricting to a trivialization patch..... Oh okay, there would be some problem because we don't know whether the restrictions are legal. I'm not an expert but I guess Tu implicitly requires that the restrictions to be legal, by talking about equivariance.

5. I'm expecting something like, for the action $\mu: P \times G \to P$, we get that

   • 5.1. $\mu(P_x \times G) \subseteq P_x$ and $\mu(P_U \times G) \subseteq P_U$ such that we can define, respectively, maps $\mu_x: P_x \times G \to P_x$ and $\mu_U: P_U \times G \to P_U$. These turn out to be actions, probably smooth actions.

   • 5.2. Each $\mu_x$ in (5.1) is transitive. (Well, this is what Proposition 27.6 says.)

   • 5.3. $\zeta_U = \mu_U$: Each $\mu_U$ in (5.1) is the action $\zeta_U$ used to determine whether or not $\varphi_U$ is $G$-equivariant

Questions:

1. Is this definition of principal $G$-bundle missing some details, such as any notion (explicit or implicit) of fiber-preservation of the action $\mu: P \times G \to P$ or any explicit description of the actions $\zeta_U: P_U \times G \to P_U$?

   • 1.1 Edit: Or any explicit mention of the relationship between $\zeta_U$'s and $\mu$

   • 1.2 Edit: Or mention of some kind of proposition that tells us the $\zeta_U$'s, which may or may not be related to $\mu$, are unique provided $\phi_U$ equivariant and $\sigma_U$ given as such

2. If the definition is in fact not missing any (explicit or implicit) notion of fiber-preserving (Edit: fiber-preserving or trivializing-open-subset-preserving) of the action $\mu: P \times G \to P$ because we can somehow deduce some kind of notion of fiber-preserving (Edit: fiber-preserving or trivializing-open-subset-preserving) of the action $\mu$ or that any of (5.1),(5.2) or (5.3) is true, then which are true, and how do we deduce these?

3. Are $\zeta_U$ and $\sigma_U$ necessarily smooth based on Tu's definition (as stated)? If not, then, based on other definitions of (smooth) principal $G$-bundle that you know, are $\zeta_U$ and $\sigma_U$ likely intended to be smooth?

   • I think I was able to prove $\sigma_U$'s are smooth by writing each $\sigma_U$ as a combination of maps, by compositions and multiplication of maps, where the maps include various projection maps and the law of composition on the Lie group $G$.

4. To clarify, the $\sigma_U$'s are free and transitive right? I think this follows from what I believe is the freedom and transitivity of the left multiplication group action of any group on itself based on its law of composition.

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Update: Can we just omit $\mu$ in the definition and then just later make a proposition about $\mu$ in the following way?

I'm thinking we instead first define that for each $U \in \mathfrak U$, $G$ acts on $U \times G$ on the right, still by the given $\sigma_U$ and then we say that $G$ acts on $\pi^{-1}(U)$ by some smooth right action $\zeta_U$ (I guess we don't have to include free or transitive since $\sigma_U$ is free and transitive and then freedom and transitivity are preserved under bijective equivariant or whatever), where $\zeta_U$

1. satisfies some compatibility condition like $\zeta_U|_{U \cap V} = \zeta_V|_{U \cap V}$ for all $V \in \mathfrak V$

2. makes $\phi_U$ is $G$-equivariant.

Later, we can make a propositions

• Lemma A. $\phi_U$ is $G$-equivariant if and only if the $\zeta_U$ is given by $$\zeta_U(e,g) = \phi_U^{-1}(\sigma_U(\phi_U(e),g)) = \phi_U^{-1} \circ \sigma_U \circ ([\phi_U \circ \alpha_U] \times \beta_U) \circ (e,g), \tag{A*}$$ where $\alpha_U: \pi^{-1}(U) \times G \to \pi^{-1}(U)$ and $\beta_U: \pi^{-1}(U) \times G \to G$ are projection maps. (In this case, I guess $\alpha_U$ is the smooth trivial action by $G$ on $\pi^{-1}(U)$.)

  • Exercise A.i. Check that $\zeta_U$ in $(A*)$ is a smooth, right, free and transitive action by $G$ on $\pi^{-1}(U)$.

  • Exercise A.ii. Check that $\zeta_U$ in $(A*)$ satisfies the above compatibility condition.

  • Equivalent Definition A.1. We use Lemma A, Exercise A.i and Exercise A.ii to say instead that $\zeta_U$ is given by ($A*$).

• Theorem B. $G$ globally acts on $P$ by some (smooth) right, free and transitive global action $\mu$ that turns out to be from collecting all the local actions, the $\zeta_U$'s, together: $\mu(p,g):=\zeta_U(p,g)$ for $p \in \pi^{-1}(U)$ for any $U \in \mathfrak U$, which is well-defined either by the compatibility condition assumption on $\zeta_U$'s in the original definition, where we don't yet know the formula for $\zeta_U$ or by Exercise A.ii, if we use $\zeta_U$ given by ($A*$).

• Corollary C1. $\mu$ is trivializing-open-subset-preserving, i.e. $\mu((U \times G) \times G) \subseteq U \times G$

• Corollary C2. $\mu$ is fiber-preserving, i.e. $\mu((x \times G) \times G) \subseteq x \times G$

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Bounty message: I really believe there's at least one of the following here:

1. ambiguity or implicit relationship between $\mu$ and $\zeta_U$'s,

2. implicit rule about uniqueness or existence of an action (in this case $\zeta_U$'s) on domain of a map that makes a map equivariant given an action (the $\sigma_U$'s) on the range

3. circular reasoning or circular definitions or something that need to be remedied either by some assumption $\mu$ preserves fibers or trivializing open subsets or by first defining smooth compatible local actions, the $\zeta_U$'s on the $P_U$'s, that make $\phi_U$'s equivariant and then later deducing a global action $\mu$ on $P$

Answer 1

I think Tu's definition is equivalent to the one involving free transitive actions on the fibers. His definition of principal $G$-bundle has two parts:

A) we have a fiber bundle $\pi:P\rightarrow M$ with $G$ acting smoothly freely on $P$ AND

B) we are told something more about the action: the fiber-preserving local trivializations are $G$-equivariant, where the action of $G$ on $U\times G$ is given by $(x,h)\cdot g = (x,hg)$.

By declaring $\phi_U$ to be $G$-equivariant, it now follows that the $G$ action on $P$ preserves fibers as follows. Suppose $g\in G$, $p\in P$ and $\pi(p)\in U\subseteq M$ where $U$ trivializes $P$. Set $\phi_U(p) = (\pi(p), h)$ and set $\phi_U(pg) = (\pi(pg), h')$. Then $$(\pi(pg), h') = \phi_U(pg) =\phi_U(p)g =(\pi(p),h)g = (\pi(p), hg),$$ from which it follows that $\pi(pg) = \pi(p)$. That is, $pg$ and $p$ are in the same fiber.

Now, to actually answer your questions:

1. The action isn't explicitly given because it's a general definition. Kind of like when defining groups, you just have some binary operation satisfying some properties. Fiber preservation was handled above.

2. Both 5.1 and 5.2 and true. I'm a little hazy on what 5.3 is asserting. But the point is the action must look like right multiplication in any trivializing open set.

3. They are smooth as they are the restrictions of the $G$ action on $P$ to a preserved subset, and the $G$ action on $P$ is assumed to be smooth.

4. Yes. (As a corollary, Tu didn't need to include "free" in his definition of prinicpal $G$-bundle, since it follows from B above.