I've finally gotten around to learning about principal $G$-bundles.

In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether these definitions are equivalent or not. I would appreciate any clarification.

All maps and group actions are assumed continuous.

Definition 1: A principal $G$-bundle is a fiber bundle $F \to P \xrightarrow{\pi} X$ together with a right action of $G$ on $P$ such that:

(1) $G$ acts freely and transitively on fibers.

(2A) $G$ preserves fibers.

Definition 2: A principal $G$-bundle is a fiber bundle $F \to P \xrightarrow{\pi} X$ together with a left action of $G$ on $F$ (note $F$ here) such that:

(1) $G$ acts freely and transitively on $F$.

(2B) There exists a trivializing cover with $G$-valued transition maps.

Definition 3: A principal $G$-bundle is a fiber bundle $F \to P \xrightarrow{\pi} X$ together with a right action of $G$ on $P$ such that:

(1') $G$ acts freely on $P$ and $X = P/G$ and $\pi\colon P \to X$ is $p \mapsto [p]$.

(2C) There exists a trivializing cover that is $G$-equivariant.

Definition 4: A principal $G$-bundle is a fiber bundle $F \to P \xrightarrow{\pi} X$ together with a right action of $G$ on $P$ such that:

(2A) $G$ preserves fibers.

(2C) There exists a trivializing cover that is $G$-equivariant.

Thoughts: It seems to me that Definition 4 is not equivalent to the other three. More than anything else, I am unclear as to why the existence of a trivializing cover that is $G$-equivariant is equivalent (is it?) to the existence of one that has $G$-valued transition functions.

I've also seen a fifth definition which assumes only condition (1).

Thanks in advance.

  • 5
    $\begingroup$ There is another one: a principal $G$-bundle is a fiber bundle with fiber $G$ and structure group $G$ where $G$ acts on itself by left translations. This is the definition of Davis & Kirk. $\endgroup$ – Bruno Stonek Aug 13 '15 at 10:31
  • $\begingroup$ @BrunoStonek, can you please help me with this definition of principle bundle (bounty offered)? I don't think there's any notion of structure group of this volume of Tu's books, though Bott and Tu of course discuss structure groups in another volume. $\endgroup$ – user636532 Oct 22 at 7:22

For the equivalence of these definitions, I would look here: Local triviality of principal bundles.

The existence of a $G$-equivariant cover is equivalent to the existence of $G$-valued transition functions:

Suppose $(U_\alpha,\Phi_\alpha)$, $\Phi_\alpha : P\vert_{U_\alpha} \to U_\alpha\times F$, is a trivializing cover. This defines a collection of maps $\phi_\alpha : P\to F$ by $$ \Phi_\alpha(p) = (\pi(p), \phi_\alpha(p)). $$ For a right principal $G$-bundle, this covering is $G$-equivariant if $\phi_\alpha(pg) = \phi_\alpha(p)g$. Now we have $$ \Phi_\alpha \circ \Phi_\beta^{-1} : U_\alpha \cap U_\beta \times F \to U_\alpha \cap U_\beta \times F $$ is an isomorphism of trivial $G$-bundles and so takes the form $$ (x, f) \mapsto (x, h_{\alpha\beta}(x,f)). $$ If the covering is $G$-equivariant then so is this map, which means that $h_{\alpha\beta}(x,fg) = h_{\alpha\beta}(x,f)g$. Since $G$ is acting freely and transitively, fixing a point of $F$ identities $F$ with $G$ and $h_{\alpha\beta}$ is entirely determined by the function $g_{\alpha\beta}: U_\alpha\cap U_\beta \to G, x \mapsto h_{\alpha\beta}(x,e)$. Thus the transition functions are given by left-multiplication by $g_{\alpha\beta}$. This is what is meant by the transition functions being $G$-valued.

Conversely, if the transition functions are $G$-valued then the trivializations will be $G$-equivariant. This is because $$ P = \sqcup_\alpha U_\alpha \times F/\sim, ~~ (x, f) \sim (x, g_{\alpha\beta}(x)f) \text{ for } x \in U_\alpha\cap U_\beta. $$ The equivariance then comes from the fact that the transition functions are operating by left-multiplication, while the $G$-action is right multiplication.

  • $\begingroup$ Hi, Eric. Thanks so much for your answer; it's very helpful. Could you perhaps clarify one more thing for me? Definition 1 is actually Wikipedia's definition. However, I don't understand why condition (2A) ($G$ preserves fibers) is included... Isn't it redundant? $\endgroup$ – Jesse Madnick May 23 '13 at 5:52
  • $\begingroup$ @JesseMadnick I actually meant to make the comment that (2A) is redundant. $\endgroup$ – Eric O. Korman May 23 '13 at 13:59
  • 3
    $\begingroup$ I have one last question, which has been bugging me most of all: Does Definition (1) imply the existence of a trivializing cover with $G$-valued transition maps? $\endgroup$ – Jesse Madnick Nov 9 '13 at 2:21
  • $\begingroup$ Definition (1) does imply the existence of a trivializing cover if the Lie group $G$ is compact, because then the projection map $\pi$ is a proper submersion from $P$ onto the base space $X$ and you can use Ehresmann's fibration theorem to assert the existence of a local trivialization. Without compactness of the Lie group, I don't think the definitions are equivalent. $\endgroup$ – Idempotent Dec 22 '15 at 12:55
  • $\begingroup$ Eric O. Korman and @JesseMadnick, can you please help me with this definition of principle bundle (bounty offered)? It might be missing some things like Eric O. Korman's statement on equivalence, though Jason DeVito's ideas in comments seem similar to the ones in Eric O. Korman's answer $\endgroup$ – user636532 Oct 22 at 7:20

In fact, these definitions are not equivalent and are not equivalent to the usual notion of a principal $G$-bundle, see e.g. Kobayashi-Nomizu "Foundations of differential geometry", Vol. I, p. 50:

First of all, you have to assume, say, properness of the $G$-action and local compactness of $F$ in all the definitions. Otherwise, the following will be a counter-example to all four: Start with your favorite connected Lie group $G$ of dimension $>0$ (say, $U(1)$) and your favorite topological space $X$ (say, a point). Then $P=G\times X$ is a principal $G$-bundle. Now, consider the same group $G$ but equipped with discrete topology $G^\delta$, but keep the original topology on $P$. Take the obvious action $G^\delta\times P\to P$. This action satisfies (1)---(4) but does not define a $G^\delta$-principal bundle.

This can be (partly) remedied by assuming that $G$ is (2nd countable!) Lie group and $F$ is a manifold. Then (2) and (3) become equivalent to the standard definition.

Here is the situation assuming the extra assumption of properness.

(1) is not equivalent to (2) even if $G$ is a compact metrizable group, see here. Nevertheless, (1) $\iff$ (2) if (in (1)) $G$ is assumed to be a Lie group ($F$ need not be a manifold; this theorem is due to R.Palais).

(2) is equivalent to (3).

(3) is equivalent to (4) provided that in (4) the $G$-action on each fiber is transitive.

  • $\begingroup$ Moishe Kohan, can you please help me with this definition of principle bundle (bounty offered)? It might be missing "proper" and possibly a lot of other things. $\endgroup$ – user636532 Oct 22 at 7:19
  • 2
    $\begingroup$ @SeleneAuckland: To be honest, I have no desire to read Tu's book since he has a bad habit of giving nonstandard definitions which are either inequivalent to standard ones or he does not bother to verify the equivalence. Furthermore, Jason knows this staff as well as anybody and is quite capable to answering the question. Lastly, if I were you, I would just use the standard definition instead of Tu's. $\endgroup$ – Moishe Kohan Oct 22 at 7:44
  • $\begingroup$ Wow. That's deep. Thanks so much for saying that. I (kinda) thought the problem was with me instead of the book. Just for the record, the issue here is 1. The definition is not really nonstandard, inequivalent or unproven but more of ill-defined or ambiguous (which worse than those 3 of course) 2. Jason seems to think Tu's definition is not ill-defined or ambiguous. $\endgroup$ – user636532 Oct 22 at 8:35
  • $\begingroup$ Anyway, are you referring to my previous question specifically, or have you encountered other questions on any of Tu's books? Also, what books on differential geometry do you like as alternatives to Connections, Curvature, and Characteristic Classes or Differential Forms in Algebraic Topology ? $\endgroup$ – user636532 Oct 22 at 8:37

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