1
$\begingroup$

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).

I believe there's something wrong with the definition of principal bundle.

Firstly, I would like to make the following points. My questions are given at the end of this post, and these points provide the context for the questions. These points are my assumptions and observations.

  1. I believe the book has no explicit definition for an action 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.

  2. Anyway, for a set $S$ (not necessarily manifold) and a group $H$ (not necessarily Lie group) and a right action of $H$ on $S$ (not necessarily free), $\alpha: S \times H \to S$, I assume the definition that $\alpha$ is "transitive" is the one on Wikipedia and assume this is equivalent to "$S$ is non-empty, and for each $s \in S$, the map $\alpha_s : H \to S, \alpha_s(h) = \mu(s,h)$, is surjective".

  3. Tu's definition of principal $G$-bundle doesn't say anything (explicitly) about $\mu$ to be transitive, fiber-preserving or anything of the sort, but whichever are true might be deduced somehow deduced from Tu's definition. 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$, $\sigma_U((x,h),g)=(x,hg)$, there is no mention of what the action $\zeta_U: P_U \times G \to P_U$ is, where $P_U := \pi^{-1}(U)$. 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 $G$-equivariant." 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, for each $U \in \mathfrak U$, the action $\mu$ on $P$ induces actions $\zeta_U$ on $P_U$.

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

    • 5.1. $\mu(P_x \times G) \subseteq P_x$ (fiber-preserving) and $\mu(P_U \times G) \subseteq P_U$ (trivializing-open-subset-preserving) 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 $\phi_U$ is $G$-equivariant

  6. I really believe there's at least one of the following here:

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

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

    • 6.3. 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 $P$


Question 1: Is this definition of principal $G$-bundle missing some details, such as any of the following?

  • 1.A. Any notion (explicit or implicit) of fiber-preservation or trivializing-open-subset-preservation of the action $\mu: P \times G \to P$

  • 1.B. Any explicit mention of the relationship between the actions $\zeta_U: P_U \times G \to P_U$ and the action $\mu$

  • 1.C. Any 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 $\phi_U$ bijective equivariant and $\sigma_U$ given as such)

Question 2: If not then, how do we deduce any of (1.A), (1.B), (1.C) ?

Question 3: Which of (5.1),(5.2) or (5.3) are true, and why?

Question 4: 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 $\zeta_U$

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

  • ii. 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 (regular) 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

  • Corollary C2. $\mu$ is fiber-preserving

Update: Question 5: What is the relationship between the collections $\{Orbit(p)\}_{p \in P}$ of orbits and $\{P_x\}_{x \in M}$ of fibers, and do these two help at all for any the previous questions? I believe that a certain relationship, described below, would imply that Tu's definition is not missing fiber-preserving because fiber-preserving can be deduced. The following bullet points provide the context for question 5 and are part of effort toward answering Question 5 and are part of effort toward answering the previous questions.

  • 5.A. I actually notice that each of the collections is a partition of $P$.

  • 5.B. I read on Wikipedia "Every orbit is an invariant subset of X on which G acts transitively." Ignoring the invariant part, I understand this to mean that for any right action, free or not, transitive or not, $\mu: P \times G \to P$ (in general, not necessarily for the one in the definition of principal bundle. Here, $P$ is just a set, and $G$ is just a group), we have that for each $x \in P$, $\mu(Orbit(x) \times G) \subseteq Orbit(x)$ such that we can define a map, which might be an action, $\mu_{Orbit(x)}: Orbit(x) \times G \to Orbit(x)$ that turns out indeed to be a right action on $Orbit(x)$ that is additionally transitive. Assuming I understand this correctly, I was able to prove this.

  • 5.C. Based on the Willie Wong's comment and on (5.A) and (5.B), it seems there's some relationship between (now going back to defining a principal bundle) $Orbit(p)$ and $P_x$ for $\pi(p)=x \in M$ or something like that. The following is all I can think of:

    • 5.C.1. I notice $p \in Orbit(p) \cap P_x$. Whenever 2 elements of the same partition intersect, they are equal. However, $Orbit(p)$ and $P_x$ belong to different partitions...Well, I assume the partitions are different. They could turn out to be equal.

    • 5.C.2. Perhaps $\mu$ free implies that $Orbit(a)=Orbit(b)$ if or only if $\pi(a)=\pi(b)$.

    • 5.C.3. For each $x \in M$ and for each $y \in P_x$, we have a map $\mu_y: G \to P$, $\mu_y(g) := \mu(y,g)$, with image $im(\mu_y) = \mu_y(G)$ $=Orbit(y)$. Each $\mu_y$ turns out to be injective because $\mu$ is free.

    • 5.C.4. For each $x \in M$ and for each $y \in P_x$, the set inclusion $\mu(Orbit(y) \times G) \subseteq Orbit(y)$. This set inclusion allows us to define a map $\mu_{Orbit(y)}: Orbit(y) \times G \to Orbit(y)$, $\mu_{Orbit(y)}(x,g) := \mu(x,g)$. By (5.B), each $\mu_{Orbit(y)}$ is a transitive right action.

      • 5.C.4.1. If we somehow have that $Orbit(y)=P_x$ for each $y \in P_x$, then the set inclusion $\mu(Orbit(y) \times G) \subseteq Orbit(y)$ is precisely $\mu(P_x \times G) \subseteq P_x$, which shows $\mu$ is fiber-preserving. I guess that $\mu$ is free is supposed to somehow show that $Orbit(y)=P_x$.
$\endgroup$
21
  • 1
    $\begingroup$ I didn't read your whole question, just the beginning. But for point 1, doesn't Tu say that the trivialization maps are $G$-equivariant? This means that $G$ is fibre preserving, no? For point 2: no, the map $\mu_x: G \to M$ is in general not surjective. Transitivity means that $\mu_x: G \to P_x$ is surjective. The fact that in the local trivialization $G$ acts on the fibre (which is identified with $G$) by right multiplication implies that $G$ must act transitively on fibres. (A group acts transitively on itself.) $\endgroup$ Oct 24, 2019 at 2:02
  • 2
    $\begingroup$ WHy can't you ask one question at a time? :O Please :( After you name the books, you start suddenly with "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." What? :O It would be easier to get the setup if you can start by saying what you want to talk about. $\endgroup$ Oct 25, 2019 at 6:31
  • 1
    $\begingroup$ If I understand correctly, in the first part of the question, you are asking if the action of $G$ on $P$ in the set up of a principal $G$-bundle $\pi:P\rightarrow M$ is transitive or not... My answer is NO... it is not true (according to any definition) that the action is transitive... It only means the action is transitive on the fibres of the map $\pi:P\rightarrow M$; that is, for each $x\in M$, the action of $G$ on $\pi^{-1}(x)$ is transitive... Is this what you are asking? $\endgroup$ Oct 25, 2019 at 6:36
  • 2
    $\begingroup$ Thanks for taking care in clarifying the concern... Edited version looks much better and cleaner... I will respond accordingly.. $\endgroup$ Oct 25, 2019 at 7:25
  • 2
    $\begingroup$ I somehow forgot about this.. It's 12:20 AM here... The first thing i would do tomorrow is to respond here... :) :) $\endgroup$ Nov 8, 2019 at 18:54

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.