Questions tagged [principal-bundles]

A principal $G$-bundle, where $G$ denotes any topological group, is a fiber bundle $\pi :P → X$ together with a continuous right action $P × G → P$ such that $G$ preserves the fibers of $P$ and acts freely and transitively on them.

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0answers
34 views

Local coordinates of one form on a principal bundle

I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates. Let's say ...
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Reference for mathematical Palatini formalism of general relativity

I know that this is maybe not a research level question, but since the topic is quite special, I thought that the chance to get some reference is higher in this community. (I already asked this ...
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Principal bundle over associated bundle

Let $P$ be a principal $G$ bundle. Let $S$ be a space with left action of $G$, and let $Q$ be a principal $H$ bundle over $S$ with the property that the action of $G$ can be lifted to $Q$. Then $$ P \...
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$1$-cocycle associated to universal $G$-bundle $EG \to BG$

Let $G$ be a (topological) group whose identity element $e_G$ is a nondegenerated basepoint (e.g. if $G$ is a Lie group). Then that's a known fact that there is for every 'nice' enough topological ...
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About irreducible connection

The irreducible connection is a connection whose holonomy group is just $G$ (let us just assume the base space $X$ is just connected). Otherwise, it is called reducible if the holonomy group $H_A$ ...
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The notion of a “relatively” flat connection

Suppose that $X$ is a connected smooth manifold and $\Gamma$ is a group acting smoothly, freely, properly and discretely on $X$, so that $Y=X/\Gamma$ is another smooth manifold endowed with a covering ...
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For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?

Title. For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$? If $G$ is a subgroup of either $S^0,S^1,S^3$ or $S^7$ this induces a free action ...
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1answer
141 views

Principal G-bundles over the circle

To edify my understanding of fiber bundles with structure groups, I was currently trying to reconcile two classifications (in a particular case). For simplicity, I'm taking the base to be $S^1$ and ...
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Dot product of functions on cosets

Some time ago I asked this same question at Math Stackexchange, because I thought that the question is nearly elementary. To my surprise, it was never answered. So I am elevating it to MathOverflow. I ...
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Relative Thom isomorphism

Let $\tilde{X}$ be a space with an action of the symmetric group $\mathfrak{S}_k$ and define $X:=\tilde{X}/\mathfrak{S}_k$ to be the quotient. On the other hand, $\mathfrak{S}_k$ acts on $(\mathbb{R}^...
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1answer
141 views

Curvature as infinitesimal holonomy 2

This question may be seen as a follow up of this original question. I'm learning Cheeger-Simons differential characters (reading Differential Characters of Bär and Becker). If I understand correctly, ...
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1answer
112 views

Is the irreducible locus of the character variety a principal bundle in Zariski topology?

Let $\Sigma$ be a compact orientable surface and let $G$ be a reductive algebraic group (say, $G=\mathrm{SL}_n(\mathbb{C})$ for simplicity). The representation variety is $$ X_G(\Sigma) = \mathrm{Hom}(...
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Milnor's universal bundle as a colimit?

I have had Milnor's construction of the classifying space of a topological group explained to me on multiple occasions, and seen it described briefly in various places. But only now am I reading the ...
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Classifying space induces a equivalence of categories between $\operatorname{PBun}_G(M)$ and $\Pi(M,BG)$ for finite groups $G$

Let $G$ be a finite group, $BG$ its classifying space and M a manifold. Then it is mentioned in Schweigert and Woike - Orbifold construction for topological field theories (Remark 2.3 d) that there is ...
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On the construction of principal $S^1$-bundles with prescribed characteristic form

I am trying to understand an argument made by Kobayashi (https://projecteuclid.org/download/pdf_1/euclid.tmj/1178245006, page 35) in his construction of a principal $S^1$-bundle with connection $1$-...
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On prequantization bundles over integral symplectic manifolds

I am trying to clarify certain subtleties regarding prequantization bundles over symplectic manifolds, for which I haven't found any clear explanation so far. Let me fix some definitions first. ...
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Evaluating the Euler class of a circle bundle on fibers

I am trying to understand what kind on information the Euler class provides about certain submanifolds of a given circle bundle. This might be completely obvious, but I don't see how to answer the ...
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Definition of universal bundle in category of smooth manifolds

Let $(P, M, G)$ a smooth principal bundle over smooth manifold $M$ with Lie group $G$. We denote by $k_{G}(M)$ the set of all classes of principal G-bundle isomorphisms. If i want to define universal ...
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Differences between induced vector fields on a smooth manifold and on a principal bundle

In the context of the connections on fibre bundle, I have found some difficulties trying to understand the fundamental vector field (my reference is Nakahara, but I'm having some problems with the ...
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The data of bundle structure of $\operatorname{End}_{0}E$ with Lie algebra

This may be a trivial question. I want to show the following claim about the structure of Lie algebra bundles, Let $X$ be a smooth projective curve over $\mathbb{C}$, and $E$ be vector bundle over $X$...
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The isomorphism classes of lifts of a $\operatorname{PGL}_r$-bundle to $\operatorname{GL}_r$

I want to show the following Lemma, The set of algebraic isomorphism classes of lifts to $\operatorname{GL}_r$ of an algebraic principal $\operatorname{PGL}_r$ -bundle $P$ on a smooth projective ...
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Lifting a metric on a principal bundle with non-compact fibre

Let $\pi \colon P \to M$ be a $G$-principal bundle. If $G$ is compact, we may lift any metric $m$ on $M$ to a $G$-invariant metric $\bar{m}$ on $P$ such that $\pi_\ast\bar{m} = m$. Doing so accounts ...
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Principal bundles over marked Riemann surface $\mathcal{M}_{g,h,r,s}$

Let $\mathcal{M}_{g,h,r,s}$ be a Riemann surface with genus $g$, $h$ boundary components, r interior marked points, and $s$ marked points on the boundary $\partial \mathcal{M} = \Sigma$. In the case ...
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Mednykh's formula for 2d manifold with boundaries? How to count principal G-bundles with prescribed holonomies?

Let S be a closed, orientable 2d manifold and G a finite group. Since a principal G-bundle over S is specified by maps $\phi : \pi_1(S) \rightarrow G$ modulo the adjoint action by G, the way to count ...
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When are principal bundles supporting Cartan connections isomorphic?

Suppose I have two Cartan geometries $(\mathscr{G}_1,\omega_1)$ and $(\mathscr{G}_2,\omega_2)$ of type $(G,H)$ over the same manifold $M$. What conditions on $G$ and $H$ allow us to conclude that $\...
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Free abelian group on a space and fibrations

Let $X$ be a topological space. Endow the free abelian group on $X$, $\mathbb Z[X]$, the quotient topology coming from the surjection $\bigsqcup_n X^n \times \mathbb Z^n \to \mathbb Z[X]$. For $Y$ a ...
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1answer
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What is the natural Lie groupoid structure on the Atiyah Lie groupoid of a principal $G$-bundle?

$\DeclareMathOperator\At{At}\DeclareMathOperator\Obj{Obj}\DeclareMathOperator\Mor{Mor}$According to https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea the Atiyah Lie groupoid $\At(P)$ of a ...
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Reference request: boundedness for semistable principal bundles on a family of curves

We work over an algebraically closed field $k$. Let $G$ be a reductive group and $X$ be a smooth projective curve over $k$. It is proven in [1, Theorem 1.2] that the moduli of semi-stable principal $G$...
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3answers
637 views

A nontrivial principal bundle which satisfies Leray-Hirsch theorem

What is an example of a nontrivial principal bundle whose fibre space $G$, total space $P$ and base space $M$ are compact connected manifolds (the fiber $G$ is a compact Lie group) such that $$H^*(P,\...
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2answers
195 views

Integrability condition for flat connections

I am reading Kobayashi's book "Differential geometry of complex vector bundles". More precisely, I am on section 2 of chapter 1, page 5. Kobayashi is trying to prove that if $E$ is a vector ...
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1answer
269 views

When a free action gives rise to a $G$-principal bundle

When a free action gives rise to a $G$-principal bundle Let a (topological) group $G$ act freely on a (topological) space $X$. Assume that $G \backslash X$ is Hausdorff. (equivalently the image of ...
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1answer
124 views

Extrinsic horizontal path lifting

As a follow up question to my previous question about the orthonormal frame bundle, I would like to understand a simple example explicitly. Let $\mathbb{S}^2$ be written extrinsically as $$\mathbb{S}...
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1answer
147 views

Čech cocycles and monodromy

It is well known that over a topological space $X$ (and choosing an open cover $\mathfrak{U}$) every locally constant Cech cocycle $g$ on $\mathfrak{U}$ with coefficients in a group $G$ yields a $G$-...
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1answer
210 views

Characterisation of (integrable) connections on (trivial) principal bundle

Let $M$ be a manifold. Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Let $P(M,G)$ be a principal bundle. Recall that, a connection on $P(M,G)$ is a distribution $\mathcal{H}\subseteq ...
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Confusion in understanding the notion of $G$ Principal bundle where $G$ is a geometric group over a site

The first paragraph of the section Overview in the paper Principal infinity-bundles - General theory by Nikolaus, Schreiber and Stevenson https://arxiv.org/abs/1207.0248 precisely reads the following: ...
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What is the geometric description of the set of isomorphism class of $G$-torsors over a site $C$?

Let $X$ be a topological space and $G$ be a topological group. Let $\tilde{G}$ be the sheaf of groups defined by the sheaf of sections of the product $G$ bundle $ \pi_1:X\times G \rightarrow X $. (1)...
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Realization of a $\mathfrak{g}$-valued $2$-form as a curvature form

Consider the Lie group $S^1$. Recall that the associated Lie algebra is $\mathbb{R}$. Let $M$ be a manifold. Consider the second de-Rham cohomology group $H^2(M,\mathbb{R})$. Let $\Omega\in H^2(M,\...
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Principal circle bundles that are smooth foliated

Let $\xi=(\pi,E,B)$ be an orientable circle bundle, i.e., a bundle with fiber $\mathbb{S}^1$ and structural group $G=\textit{Diff}^+(\mathbb{S}^1)$. Claim 1: The bundle $\xi$ admits a structure of $...
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1answer
498 views

Some questions about Hitchin's self-duality paper

I am reading this paper (The self-duality equations on a Riemann surface by N. Hitchin), and I don't understand a few things in page 67. In proof of Theorem 2.1 after Equation 2.4, he gives the ...
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Does holonomy determine parallel transport? [duplicate]

Let $p: P \longrightarrow M$ be a smooth $G$-principal bundle endowed with a connection that determines the holonomies: $$\Phi_{\gamma}: P_{x} \overset{\cong}{\longrightarrow} P_{x}$$ for any fiber ...
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119 views

Diagonal action on external product of trivial principal bundles

(Title, tags and phrasing of this problem might not very good, so please feel free to edit it.) In the course of writing a long and technical proof, I recently came across the following problem: Let ...
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Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates

In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented. Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...
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1answer
76 views

Holomorphic local trivialization of a principal toric bundle

Let $G$ be an even-dimensional compact Lie group with Lie algebra $\mathfrak{g}$ and let $T \subset G$ be a maximal torus with Lie algebra $\mathfrak{t}$. We can construct a left-invariant complex ...
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1answer
169 views

basic question on quotient stacks

Let $X$ be a scheme over $S$, and $G$ be an affine group scheme over $S$ acting on $X$. This Wikipedia article (or also this related MO question) defines a quotient stack $[X/G]$ as a category of ...
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2answers
298 views

Fibration of principal bundles

Let $G$ be a topological group, let $f:X\rightarrow Z$ be a $G$-equivariant map of (left) $G$-spaces such that $X\rightarrow X/G$ and $Z\rightarrow Z/G$ are principal $G$-bundles. $f$ is a ...
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1answer
204 views

Chern -Weil map for topological principal G bundles

Let $G$ be a Lie group. In the book Curvature and Characteristic classes, the author (Johan L. Dupont) mentiones in beginning of chapter 5 the following : The notion of a topological principal $G$...
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Poincaré connection encode torsion and curvature

I'm trying to understand something that is written in Baez & Wise paper "Teleparallel Gravity as a Higher Gauge Theory". In section 4, they discuss Poincaré connection and, first of all, split ...
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3answers
813 views

An intuitive explanation for group cohomology via cochains?

I'm fairly new to topology, and so far I've understood cohomology via cochains. First we build an object called a cochain ($C^n$), then define a differential map that takes you from $C^n$ to $C^{n+1}...
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241 views

Is there something wrong with this definition of principal bundle?

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 (...
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180 views

Introducing connection on principal bundle as lifting of vector field and paths

Let $\pi:P\rightarrow M$ is a principal $G$ bundle. I want to introduce the notion of connection as a way to uniquely lift the structures on $M$ to structures on $P$, namely vector fields and paths. ...

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