In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product $X\times G$ of a space $X$ with a group $G$.

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479 views

Local triviality of principal bundles

Suppose I define a principal $G$-bundle as a map $\pi: P \to M$ with a smooth right action of $G$ on $P$ that acts freely and transitively on the fibers of $\pi$. Does it follow that $P$ is locally ...
10
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1answer
232 views

Why base point makes a huge difference?

While preparing a talk, I was tempted to "prove" the following relationship: $$\text{Prin}_{G}(X)\cong [X,BG]\cong [B[\pi_{1}(X),BG]\cong [\Omega B[\pi_{1}(X)],\Omega BG]\cong [\pi_{1}(X),G]$$ Here ...
9
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3answers
953 views

Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory

I am looking for a textbook that might serve as an introduction to principal bundles, curvature forms and characteristic classes, and perhaps towards 4-manifolds and gauge theory. Currently, the only ...
8
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1answer
254 views

Equivalence of Definitions of Principal $G$-bundle

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 ...
8
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1answer
75 views

What the curvature $2$-form really represents?

Let $(E,\pi,B)$ be a principal bundle with structure group $G$. The connection $1$-form can be thought of as a projection on the vertical part. It allows us to characterize the horizontal subspaces as ...
7
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205 views

Orbit space of a free, proper G-action principal bundle

Let $G$ be a topological group and let $r \colon E \times G \to E$ be a continuous right-action on a topological space $X$. If $p\colon E \to B$ is a continuous map into a topological space $B$ such ...
6
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1answer
147 views

Flat connection with non-trivial holonomy? I cannot get it

maybe this is a dumb question, but I cannot understand how a principal $G$-bundle can have non-trivial holonomy with a flat connection. Maybe I'm missing something, but doesn't Ambrose-Singer theorem ...
6
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1answer
107 views

Formal Definition of Yang Mills Lagrangian

I have a question regarding the Lagrangian in non abelian gauge theory. Say, $G$ is the gauge group and $\mathfrak g$ the associated Lie algebra. The Lagrangian is often written as $$ \mathcal ...
6
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0answers
89 views

Exact sequence of tangent spaces of principal $G$-bundles

Let $P$ be a smooth manifold, $G$ a Lie group, $\alpha:P\times G\to P$ a smooth action and $p:P\to P/G$ a smooth principal $G$-bundle. Then, we have the sequence $$ G \xrightarrow{\alpha_a} P ...
5
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1answer
159 views

How does a left group action on the fiber of a principal bundle induce a right action on the total space?

Suppose I define a "principal $G$-bundle" as follows: A principal $G$-bundle is a fiber bundle $F \to P \overset{\pi}{\to} X$ with a left group action of $G$ on $F$ that is free and transitive, ...
5
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1answer
61 views

When can a connection be lifted?

Let $P \rightarrow X$ be a principal $G$-bundle, and $P' \rightarrow X'$ be a principal $G'$-bundle. Let $(f',f'')$ be a morphism from $P'$ to $P$, i.e., a pair of maps $f': P' \rightarrow P$ and ...
5
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1answer
67 views

Intuition behind the connection one-form

When we define a connection on one principal bundle $\pi: P\to M$ with structure group $G$ we define it as an association of one subspace $H_pP\subset T_pP$ for each $p\in P$ such that $T_pP = H_pP ...
5
votes
1answer
128 views

One-form on quotient manifold

Let $M$ be a smooth manifold with tangent bundle $TM$ and cotangent bundle $TM^*$ and $\psi\in TM^*$ a one-form. We denote the quotient manifold of $M$ by the free and proper $G$-action $\varphi$ as ...
5
votes
2answers
109 views

Principal $\mathrm{SL}_n$-bundles

It seems to be well-known that a principal $\mathrm{SL}_n$-bundle on a scheme or manifold $X$ is the same as a vector bundle of rank $n$ whose determinant is a trivial line bundle. One direction is ...
5
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0answers
36 views

Cartan geometry on manifolds with boundary

I was reading Sharpe's text on Cartan geometry, and I started to wonder: Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth ...
5
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1answer
119 views

Explicit description of flat connections under pullback on principal bundles over Riemann surfaces

I'm trying to find a proof/reference for a statement that I've seen quoted in some way or the other, but without reference. The setting: let $P\longrightarrow M$ be a flat principal $G$-bundle over ...
5
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2answers
125 views

Connections on a manifold and principal connections on the frame bundle

Suppose $M$ is a manifold, and $E$ a vector bundle over $M$ equipped with a connection $\nabla $. If $F$ is the frame bundle of $E$, is there an explicit construnction of a connection on $F$ ...
4
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2answers
101 views

Classify sphere bundles over a sphere

Problem (1) Classify all $S^1$ bundles over the base manifold $S^2$. (2) Do the same question for $S^2$ bundles. Moreover, does there exist a universal method to solve this kind of ...
4
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1answer
193 views

Principle G bundles v.s. Flat G connection

What is the difference between Principle G bundles v.s. Flat G connection? I heard that for a discrete group $G$ (in physics, or a finite group $G$ in math), the principle G bundles is the same ...
4
votes
1answer
60 views

“Bundle of metrics” on a principal bundle?

I've come across the term "bundle of metrics" on a principal bundle. In particular, my setting is that for $N\longrightarrow M$ a universal cover of a compact Riemann surface, $P\longrightarrow M$ a ...
4
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1answer
452 views

Intuition of Chern-Weil theory

Let $ P \rightarrow M$ be a $G$-principal bundle. The lie algebra of $G$ is $\frak{g}$ and $P$ has connection form $\omega \in H^1(P,\frak{g})$ and curvature form $\Omega \in H^2(P,\frak{g})$. We ...
4
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1answer
125 views

Problems understanding the construction of Hitchin moduli space in his paper “The self-duality equations on a riemann surface”

First, if this post must be broken up in separate questions, please tell me so. I thought it would be better if I simply posed my questions in one thread, as they are directly related to each other. ...
4
votes
1answer
179 views

Classifying map

Let $\xi=(E,p,B)$ a principal $G$-bundle and $\eta=(P,\pi,Q)$ a real vector bundle such that $\operatorname{rank}(\eta)=n$. We can consider a classificant space $BG$. What is the classifying map $f:X ...
4
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1answer
29 views

Sections of associated bundles

Let $\pi:P\rightarrow M$ be a Principal bundle and $\pi_V:P\times_G F\rightarrow M$ be its associated bundle via the representation $\rho:G\rightarrow GL(V)$. Fact: $\Gamma(P\times_G ...
4
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2answers
83 views

Principal bundles, connection forms and fundamental vector fields

Suppose $\pi:P\rightarrow M$ is a principal bundle, $\omega\in \Omega^1(P;\mathfrak{g})$ is the connection one form and $\sigma(\cdot)$ is the fundamental vector field associated to some vector field ...
4
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1answer
58 views

Lifting Riemannian metrics on principal bundles

Given a principal bundle $\pi:M\rightarrow M/G$, there are natural maps $$\pi_{\mathcal{F}}:\mathcal{F}(M)^G\rightarrow\mathcal{F}(M/G)$$ ...
4
votes
1answer
105 views

Canonical connection on $CP^n$

I have heard something along the lines of "There is a canonical $U(1)$ connection on $CP^n$" and I am trying to understand what that means. First I suppose that the sentence refers to a line bundle ...
4
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0answers
24 views

Isomorphism of Principal Bundles with structure groupoid

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Suppose that $\pi:P\rightarrow B$ is a $\mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)\rightarrow (P,B); s\in [0,1]$, be a homotopy ...
4
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0answers
112 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
4
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0answers
91 views

Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...
4
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186 views

Group actions and associated bundles

Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
3
votes
2answers
114 views

Is the $G$-action on a principal $G$-bundle proper?

Let $G$ be a Lie group. If $G$ acts properly and freely on a manifold $P$, then it is well-known that $P \to P/G$ form a principal $G$-bundle. I would like to know the converse: namely Question: if ...
3
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1answer
93 views

Naive question: what good are characteristic classes of principal bundles?

I recently read a development of characteristic classes on principal bundles through curvature forms and the Chern–Weil homomorphism. Unfortunately, this exposition concluded without listing any ...
3
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1answer
129 views

Learning Fibre Bundle from “Topology and Geometry” by Bredon

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if ...
3
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1answer
126 views

When does the difference between a vector bundle and the associated frame bundle matter?

In the comments to this question How a principal bundle and the associated vector bundle determine each other, it was remarked that while there is a bijective correspondence between rank $n$ vector ...
3
votes
1answer
55 views

Associated bundles: isomorphism between spaces of differential forms.

I think this will be an easy question for numerous people. Let $\pi:P\rightarrow M$ be a principal bundle and $\rho:G\rightarrow GL(V)$ a representation. The space of $k$ forms on $M$ with values in ...
3
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0answers
43 views

Upgrading Leray–Hirsch to Künneth for principal bundles

The Leray–Hirsch theorem says that given a fiber bundle $F \to E \to B$ such that $H^*(F)$ is free (as a module over whatever coefficient ring $k$) and, for each $n \geq 0$ there is a set of classes ...
3
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0answers
68 views

Reference for principal bundles and related concepts

I am looking for a good reference for fibre bundles, Ehresmann connections, principal $G$-bundles and principal Ehresmann connections (the $G$-equivariant version of Ehresmann connections). Could ...
3
votes
0answers
48 views

How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...
3
votes
0answers
88 views

What is the commutator of a horizontal and vector field for a connection on a Fiber bundle?

I would be tempted to rephrase my question as : why do people seem to care only about the curvature of a connection on fiber bundles ? Indeed, the curvature gives the vertical part of the commutator ...
2
votes
1answer
347 views

How a principal bundle and the associated vector bundle determine each other

It seems to me that given a vector bundle, the associated principal bundle is univocally determined. In fact one has to construct a principal bundle given the base, the fibre (the group $G$ in which ...
2
votes
2answers
91 views

If a connection on a principal $G$-bundle restricts to an $H$-subbundle, must its holonomy lie in $H$?

Let $P \to M$ be a principal $G$-bundle, equipped with a principal connection $D$. Let $Q \subset P$ be a principal subbundle with fiber $H$, where $H \leq G$ is a (let's say closed and connected) ...
2
votes
1answer
55 views

The curvature of the connection one form - misunderstanding

Let $(P,M,G,\pi,\cdot)$ be a principal bundle. Let $\omega$ be the connection one form for a connection $H\subset TP$. Let $X,Y$ be smooth vector fields on $P$. Then the curvature $\Omega$ of the ...
2
votes
3answers
210 views

$M \times N$ orientable if and only if $M, N$ orientable

For two manifolds $M$ and $N$ I'm trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then ...
2
votes
1answer
200 views

If there exists a global section then the principal bundle is trivial - problem with smoothness

Let $\pi \colon P \to B$ be a principal $G$-bundle and let $s \colon B \to P$ be it's smooth section. In order to show that $P \simeq B \times G$ I define the map $\varphi \colon P \to B \times G$ by ...
2
votes
1answer
86 views

Characteristic classes not defined on vector bundles

If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking ...
2
votes
2answers
216 views

Principal bundle automorphism generating global gauge transformations

Consider a principal $G$-bundle $P$ with connection form $\omega$. An automorphism $f$ of $P$ is by definition a (smooth) $G$-equivariant map: $f(p \cdot g) =f(p) \cdot g$ for all $p\in P$ and $g\in ...
2
votes
1answer
153 views

Connections and covariant derivatives

Let $A$ be a connection on a principal $G$-bundle $P$, let $\chi :G\rightarrow GL(V)$ be a representation of $G$, and let $E:=P\times _\chi V$ be the associated gauge bundle. Then, there is a ...
2
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2answers
109 views

Manifold non-orientable iff. frame bundle is connected

Let $M$ be a connected smooth manifold and $L(M):=\bigcup_{x\in M}L_xM$ its frame bundle where $L_xM:=\{(v_1,\dots,v_n):\{v_1,\dots,v_n\}\text{ is a basis of }T_xM\}$. $M$ is non-orientable iff. ...
2
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2answers
129 views

Universal property of universal bundles.

A classifying space for a group $G$ is a topological space $BG$ with a principle $G$-bundle $p : EG \to BG$ where $EG$ is contractile, so that $BG = EG/G$. A classifying space is universal in the ...