Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [principal-bundles]

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 $G$.

1
vote
0answers
8 views

Characteristic classes for $P \rightarrow G \rightarrow G/P$

Let $G$ be a complex semisimple Lie group and let $P$ be a parabolic subgroup. We know that the cohomology of the flag variety $G/P$ is generated by Schubert classes. There is a principal $P$ bundle, ...
0
votes
0answers
12 views

Help find mistake in conclusion about vector fields on principal bundle

Let $P$ be a principal fiber bundle with structure group $G$ acting freely on the right. Let $T_uP = G_u + Q_u$ be a connection on $P$, where $u\in P$ and $G_u$ is the vertical space consisting of ...
0
votes
0answers
19 views

Homotopy invariance of pullbacks of principal bundles

This is the proof of lemma 7.2 in a notes by Stephen Mitchell, on classifying spaces. Essentially one step of the proof claims that: Let $p:Y \rightarrow B \times I$ be a principal bundle. If $B$...
2
votes
1answer
55 views

Literature suggestion for understanding Gauge theory from the perspective of a Mathematician.

Can anyone please suggest some good literature or references for understanding Gauge theory from the perspective of a mathematician (from the point of view of differential geometry)? Being a ...
1
vote
0answers
118 views

Exact sequence induced by an exact sequence of groups

Suppose $H\hookrightarrow G$ is an inclusion of topological groups and $H$ is closed in $G$. Then we have an exact sequence $$ 1\to H\to G\xrightarrow \pi G/H \to 1$$ where $G/H$ is just a pointed ...
1
vote
1answer
37 views

Computing the differential of a Lie group action

(Ex. 27.4 page 252 Loring Tu) (The differential of an action). Let $\mu: P \times G \rightarrow P$. For $g \in G$, the tangent space $T_gG$ may be identified with $l_{g*} \mathfrak{g} $, where $...
4
votes
1answer
42 views

Does maps between fundamental groups induces a continuous map between spaces?

Main question is this: Suppos $M$ is a manifold and $G$ is a finite group. If there is a group homomorphism $\phi:\pi_1 M\to G$, is there a continuous map $f:M\to BG$, where $BG$ is a classifying ...
0
votes
0answers
25 views

Natural bijection between equivariant maps and sections, Principal bundles

This is in page 11, Prop 6.1 of Mitchell's notes on Fibre bundles. I am quoting the proposition below. Let $\pi:P \rightarrow B$ be a principal $G$ bundle, $X$ a right $G$-space. $Hom_G(P,X)$ ...
2
votes
2answers
134 views

Doubt in the definition of principal bundle.

I am following Kobayashi and Nomizu( Foundations of differential geometry) Volume 1. In page number 50 while defining principle $G$-bundle $P(M,G)$ they said that the action of the Lie group $G$ ...
0
votes
1answer
33 views

Question about a proposition in Kobayashi book about $G$-structures.

I'm reading Kobayashi's book, Transformation Groups in Differential Geometry and at the page 3 is this proposition: the definition of $K$ is given here: My question is why this proposition is ...
1
vote
1answer
65 views

Reduced bundles and global sections of associated bundle

I am following Kobayashi and Nomizu's book Foundations of differential geometry volume 1 (page no $57$) Proposition 5.6. The structure group of a principal bundle $P(M,G)$ is reducible to a closed ...
0
votes
0answers
23 views

Local frame inducing a map of principal bundles

Let $V \rightarrow M$ a vector bundle. $P \rightarrow M$ a principal $G$-bundle. Let $\phi:G \rightarrow GL(V)$ be a representation. A local section $s$ for $P$, frame bundle for $V \rightarrow M$ , ...
0
votes
0answers
26 views

$H$ is admissible, then $P\rightarrow P/H$ is a principle $H$ -bundle.

Proposition 3.5, page 5: Suppose $P \rightarrow B$ is a principal $G$-bundle, and let $H$ be an admissible subgroup of $G$. Then the quotient map $P \rightarrow P/H$ is a principal $H$-bundle. ...
1
vote
1answer
47 views

Different notions of torsors in algebraic geometry

In what follows $X$ will be a scheme and $G$ a group scheme. In the examples I will take $X=\mathbb{P}^1_k$ and $G=\mathbb{G}_{m}$. When reading about "the torsor..." I found many definitions, not ...
0
votes
0answers
34 views

Connection $1$-form acting on vector fields

I'm reading this paper about the c-map between special Kähler manifolds and Hyperkähler manifolds and in the introduction the authors talk about the cotangent bundle as a certain associated bundle of ...
0
votes
0answers
22 views

Connection/Curvature as a matrix of Real valued forms

Let $P(M,G)$ be a principal $G$ bundle. Let $\omega$ be a connection $1$ form on $P(M,G)$. This is a $\mathfrak{g}$ valued $1$ form on $P$ i.e., for each $p\in P$, we have $\omega(p):T_pP\rightarrow ...
2
votes
2answers
36 views

What does determinant bundle of a principal bundle say about the principal bundle

Let $\pi:P\rightarrow M$ be a principal $Gl(n,\mathbb{R})$ bundle. Given $x\in M$ there is an open set $U$ containing $x$ and a local trivialization $\pi^{-1}(U)\rightarrow U\times G$. This gives a ...
2
votes
1answer
29 views

Isotropy group of connection is isomorphic to centraliser of holonomy group

I am asking for a proof of Lemma (4.2.8) of Donaldson, Kronheimer: The Geometry of Four-Manifolds. Let $P \rightarrow X$ be a principal bundle with structure group $G$. Denote by $\mathcal{G}$ the ...
2
votes
1answer
35 views

Is there a group whose manifold is a fiber bundle with base is $S_1$ and fiber $\mathbb{Z_2}?$

Let's consider a fiber bundle with base $S_1$ and fiber $\mathbb{Z}_2$. I want this manifold to be topologically non-trivial, the edge of the Möbius strip. How do I know if is it possible to ...
6
votes
0answers
131 views

Is a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle over a 3-manifold $M$ equivalent to an element in $H^1(M,\mathbb{Z}_2)\times H^2(M,\mathbb{Z}_4)$?

Given a 3-manifold $M$ and a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle $P$ over $M$ whose isomorphism class is represented by the homotopy class of a map $f:M\to B(\mathbb{Z}_2\ltimes PSU(4))$ ...
2
votes
1answer
84 views

Showing $GL_n$ is a special algebraic group

So there's this notion of a group scheme $G$ being 'special' if any principal $G$-bundle over a scheme $X$ (say defined in the etale topology) is also locally trivial in the Zariski topology. I would ...
0
votes
0answers
8 views

Universal standard principle bundle of the Gauge group

Let $p:P\rightarrow B$ be a principal $G$-bundle, and let $E_G\rightarrow B_G$ be the universal bundle for $G$. Let $Aut_B(P)$ be the subspace of $Map_G(P,P)$ of maps over $B$. Let $Map_P(B,B_G)$ be ...
0
votes
1answer
20 views

Problem about frame bundle in Kobayashi's book

I'm reading Kobayashi's book "Transformation Groups in Differential Geometry" and I have a problem in the proof of this lemma: At the converse part he says this: My question is about $f,$ namely, ...
1
vote
1answer
39 views

Principal bundles with quotient map

I'm trying to prove that if $G$ is a Lie group and $H < G$ a closed subgroup, and we have the quotient map defined as $\pi: G \rightarrow G/H$, then $(G, \pi, G/H, H)$ is a $H$-principal bundle ...
0
votes
1answer
19 views

Quick question on structure groups of the frame bundle of a Manifold

Let $M$ a manifold and $G$ a group. Is it true that the statement: "the structure group of the frame bundle of $M$ can be reduced to $G$" simply means: There is a subbundle $S$ of the frame ...
4
votes
1answer
99 views

Covariant derivative: QFT vs. Math

In class, we have seen that the covariant derivative of some form $R$ can be written as: $$DR = dR + [A, R] = dR + A\wedge R - R\wedge A \tag1$$ Here, $d$ represents the external derivative over ...
4
votes
0answers
40 views

Extend the fiber of a principal $PSU(n)$-bundle

For $n>2$, the outer automorphism group of $PSU(n)$ is $\mathbb{Z}_2$. My question: Given a principal $PSU(n)$-bundle $P$ over a manifold $M$, can we extend the fiber of $P$ to $\mathbb{Z}_2\...
0
votes
0answers
17 views

Any example of optimization on a fiber bundle?

In engineering, concepts like manifolds (with boundary) arise naturally in constrained optimization. Sometimes, the domain of optimization is naturally identified with a Lie group. Theoretically ...
1
vote
0answers
32 views

Defining a connection in $\mathbb{R}^2$ using a connection $1$-form

I'm reading Hitchin's paper Self-duality Equations on a Riemann Surface (Hitchin, self duality). In the first chapter on pages 63/64 he considers a principal $G$-bundle $P$ over $\mathbb{R}^4$ and a ...
0
votes
0answers
21 views

Fibred charts adapted to principal bundle structures

If $\pi_E:E\rightarrow M$ is a rank $k$ vector bundle (let's assume everything in this question to be real for simplicity), it is the most common to use fibred charts adapted to the vector bundle as ...
0
votes
0answers
18 views

Principal Bundle definition

Let $G$ act on smooth manifold $M$. If the action is free $Stab(G) = \{ e\}$, then $O(p) = Orbit(p) \approx G$. In the definition of principal bundle with $(E,\pi, B, G)$, then let $G$ acting on $E$ ...
0
votes
0answers
20 views

Associativity of balanced products for $G$-spaces

This is Proposition 3.1 pg 4 Let $X$ be a right $G$-space, $Y$ a $(G,H)$ space, $Z$ a left $H$ space, then there is a natural homeomorphism. $$(X \times _G Y) \times_H Z \cong X \times_G( Y \...
1
vote
1answer
44 views

principal $G$ bundle from principal $H$ bundle given a morphism of Lie groups $\phi:G\rightarrow H$

Let $\phi:G\rightarrow H$ be a morphism of Lie groups. Given a principal $G$ bundle, we can associate a principal $H$ bundle by what is called associated fiber bundle for a principal bundle. Can we ...
0
votes
1answer
31 views

changing the structure group along the given homomorphism of Lie groups

Let $P\rightarrow M$ be a principal $G$ bundle and $\phi:H\rightarrow G$ be a morphism of Lie groups. Can some one help me to understand the constrution of reducing the structrue group to $H$ i.e., ...
0
votes
1answer
40 views

Map from schemes to stacks

I have just started studying stacks. Trying to understand the theory I was thinking about a (very interesting) toy example: $ BG $ the classifying stack of a smooth (over a base scheme $ S $) group G. ...
0
votes
0answers
37 views

Transition maps of principal bundle are smooth

Let $P(M,G)$ be a principal bundle. We choose an open covering $\{U_\alpha\}$ of $M$ and trivializations $\psi_\alpha:\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times G$ defined as $\psi_\alpha(u)= (\...
2
votes
1answer
62 views

Looking at the connection 1-form on a principal G-bundle in coordinates

I'm reading this paper, and I'm confused about something. Let $A$ be a connection on a principal $G$-bundle $P$ over $\mathbb{R}^4$, and $F(A)$ its curvature. Let $\mathrm{ad}(P)=P\times_G\mathfrak{g}...
1
vote
1answer
44 views

G-principal bundle and homotopy retract

Suppose that $f:X\rightarrow Y$ a continuous map between (connected) CW-complexes such that there exists a continuous map $g:Y\rightarrow X$ with the property that $g\circ f$ is homotopy equivalent to ...
2
votes
1answer
152 views

Showing the Hopf fibration has no global sections

Let's consider a principle $U(1)$-bundle over $S^2$ with the transition function $g_{\infty 0} = z/|z|$ (it is known as the Hopf fibration). There is a simple topological argument showing that this ...
0
votes
0answers
24 views

Showing simple transivity of action on fibre

Let $(P,X,\pi, G)$ be a principal $G$-bundle (definition below). I want to show (a) $\pi(u\cdot g)= \pi (u)$ (b) If $\pi(u)=\pi(v)$ then, there exists unique $g\in G$ such that $u=v\cdot g$. (a) ...
1
vote
0answers
32 views

Why is a connection on the bundle $SO(M)$ metric compatible?

If we have an orientable manifold $M$ with a metric $g$ and signature $(r, s)$, we can define the principal-$SO(r, s)$ bundle $SO(M)$, the bundle of orthonormal frames of $TM$. This is a subset of the ...
0
votes
0answers
34 views

Transitivity of the action in definining principal bundles

In the wikipedia article https://en.wikipedia.org/wiki/Principal_bundle the definition of a principal $G-$bundle $\pi:P\rightarrow X$ demands that the action of $G$ on $P$ to be free and transitive. ...
2
votes
0answers
86 views

Correspondence between flat connections and fundamental group representations

Let $M$ be a manifold. Two stackexchange posts state that there is a correspondence $$ \{ (P,A): P \text{ a $G$-bundle}, A \text{ flat connection} \} \leftrightarrow \{ \text{morphisms } f:\pi_1(M) \...
0
votes
1answer
108 views

Universal Bundle — Understand the basic definition

I read two versions of discussions on universal bundles. I could not really see how the two definitions are really the same. From Wiki. The universal bundle in the theory of fiber bundles with ...
3
votes
0answers
45 views

Two notions of classifying space

Why is the categorical classifying space for a group G, i.e., geometric realization of the nerve of G(as a category of One object), the same as the topological classifying space for principle G ...
2
votes
0answers
68 views

Line bundles associated to principal circle bundles

Let $\pi: P \rightarrow B$ be a principal circle bundle over $B$ and $\rho: S^1 \times \mathbb{C} \rightarrow \mathbb{C}$ an effective left action. Then, one can associate to the bundle $\pi$ a ...
4
votes
1answer
85 views

Transitive action of $H^2(M;\Bbb Z)$ on $Spin^c$ structures over $M$

I’ve a problem understanding why the action of the second cohomology group (integer coefficients) of an oriented smooth manifold $M$ is free and transitive on the set of $Spin^c$. I’m following these ...
2
votes
1answer
70 views

If $H\leq G$, then $BH \to BG$ is a fiber bundle with fiber $G/H$

Suppose $G$ is a topological group and $H\leq G$ is a closed subgroup. The inclusion $H\to G$ induces a map on classifying spaces $BH\to BG$. I've seen in some sources that $BH\to BG$ is actually a ...
0
votes
0answers
37 views

Classification of principal bundles and of regular coverings

Let $G$ be a discrete group and let $X$ be a good space with fundamental group $\pi_1$. We know the following things: The connected principal $G$-bundles over $X$ are exactly the regular coverings of ...
3
votes
0answers
88 views

From sheaf torsors to geometric bundles on schemes

$\DeclareMathOperator{\Spec}{Spec}$ $\DeclareMathOperator{\Sym}{Sym}$ $\newcommand{\func}{\mathcal{O}}$ $\newcommand{\M}{\mathcal{M}}$ It is well known that the notions of locally free $\func_X$-...