# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Open problems in fiber bundles theory

As the title says, what are some problems in fiber bundles theory (especially principal bundles) that are still open?

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### Uniqueness of Witten-Dijkgraaf 2D TQFT at 0th dimension

If I understand correctly, Dijkgraaf-Witten TQFT in dimension 2 is the
following. Fix any finite group $G$, we define a field over a closed
2-manifold to be a principle $G$ bundle (it's automatically ...

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### The fixed points set of the actions of $\mathbb{C}^*$ and $S^1$ on the Higgs bundle moduli space

Let $\mathcal{M}_{d}(G)$ be the moduli space of $G$-Higgs bundles. $\mathcal{M}_{d}(G)$ have a non-trivial $\mathbb{C}^{*}$-holomorphic action by multiplication of the Higgs field,
$$
z\cdot (E, \...

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### Existence of horizontal lifts in $G$-bundles

I wanted to show that for any smooth principal $G$-bundle $E\xrightarrow\pi B$ any smooth curve $\gamma\colon I\to B$ has a unique horizontal lift from a fixed starting point $u_0\in\pi^{-1}\left(\...

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### Holonomy map on a connected manifold determines the connection and the bundle

I am reading Parallel transport on principal bundles over stacks. I quote from their paper :
Let $G$ be a Lie group and $M$ a $C^{\infty}$ manifold. Recall that a
choice of a connection $1$-form ...

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### Advantages of Atiyah sequence version of connections on a principal bundle

I am reading Lie Groupoids and Lie Algebroids in Differential Geometry
by Kirill Mackenzie.
In appendix (page $291$), before discussing about Atiyah sequence associated to a Principal bundle, the ...

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### Proof that geodesics have zero curvature with Cartan's moving frames method

I tried to use Cartan's moving frames method to prove that any minimizing-length curve in $\mathbb{R}^2$ has zero curvature. Here below is my idea of proof. I am asking for a proof-verification and ...

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### Extension of Hopf fiber bundle to (an equivariant) 2 dimensional vector bundle

Let $p:S^3 \to S^2$ be the Hopf fibration which is a result of the standard action of $S^1$ on $S^3$.
Is there a $2$ dimensional vector bundle $\tilde{p}:E \to S^2$ such that $S^3\subset E$ ...

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### Descending central extensions to homogeneous spaces

Let $G$ be a Lie group (finite dimensional or Banach), and let $H$ be a Lie subgroup (in the Banach case we assume that $H$ is a submanifold which is also a Lie group). Let $\text{U}(1) \rightarrow \...

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### Natural morphism to the scheme of isomorphism

Suppose that we have a faithful representation $\rm{G}\rightarrow\rm{GL}(V)$ of a semisimple linear algebraic group into a complex vector space $\rm{V}$ of dimension n. Suppose that we have a ...

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### Integrability of certain distribution associated to a connection form on the total space of a principal bundle (Principal Frobenius condition)

Let $P\to M$ be a $G$-principal bundle where $P,M$ are smooth manifolds and $G$ is a Lie group with Lie algebra $\mathfrak{g}$, whose center is denoted by $C(\mathfrak{g})$.
Let $\omega$ be the ...

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### Using Stiefel-Whitney class to build new principal bundles

I'm reading this paper and at the beginning of the second section, he states many results that aren't clear to me.
Consider a principal $SO(3)$-bundle $P\rightarrow R^2\times \Sigma$, where $\Sigma$ ...

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### Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach

Let $P\rightarrow M$ be a principal fibre bundle with structure group $G$, $F$ a manifold and $\alpha: G\times F\rightarrow F$ a smooth left action.
There is an associated fibre bundle $E\rightarrow ...

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### Mapping class group orbits of principal bundles

Suppose $M$ is a manifold (I would be happy with low-dimensional examples like surfaces, but let me ask more generally).
Then for any discrete group $G$ (again, I would be happy with a finite group) ...

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### Noncommutative Leray - Hirsch theorem in the context of noncommutative principal bundles

In the literature, are there some researchs on non commutative analogy of Leray-Hirsch theorem in the context of non commutative Principal bundles?

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### Projection of an invariant almost complex structure to a non-integrable one

My apologies in advance if my question is obvious or elementary.
We identify elements of $S^3$ with their quaternion representation $x_1 + x_2i + x_3j + x_4k$. We consider two independent vector ...

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### Is there a representation theoretic way to define the pullback of densities and differential forms?

I find it convenient to define the bundle of densities of weight $\alpha$,say $\Omega_\alpha(M)$ over a smooth manifold $M$ as the associated vector bundle of the frame bundle $F(M)$ with the ...

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### When is the action of the gauge group on the space of connections free?

Let $G$ be a compact Lie group. Let $\mathcal{A}$ be the space of connections on the principal trivial $G$-bundle $G\times \mathbb{R}^4$ possibly with some growth condition (to specify it is a part of ...

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### How to prove two curves in the frame bundle to project to the same curve on base manifold?

There is a problem about Cartan's development, arising from the paper 'Kinetic Brownian motion on Riemannian manifolds', Subsection 2.4.1. To be precise, let $(M,g)$ be a $d$-dimensional complete ...

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### Atiyah Sequence and Connections on a Principal Bundle

Let $G$ be a Lie group and $\pi:E_G\rightarrow M $ be a principal $G$-bundle.
I have seen in many places that a connection on $(E_G,M,G)$ is a splitting of the Atiyah sequence
$$ 0\rightarrow \text{...

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### Gauge structure of teleparallel gravity

I am interested in references that treat teleparallel gravity in a mathematically rigorous manner, especially in regards to it being a "gauge theory of the translation group".
The standard reference ...

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### Classification of bundles, Postnikov towers, obstruction theory, local coefficients

RECAP on classification of bundles
We want to classify $G$-principal bundles over $X$ (smooth manifold, G compact Lie). These are in 1-1 correspondence with homotopy classes of maps $[X,BG]$ (where $...

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### Homogeneous space for intersection of subgroups

Suppose we have a Lie group $G$ and two subgroups $P_1$ and $P_2$. We can then study the homogeneous spaces $M_1=G/P_1$ and $M_2=G/P_2$, and bundles on these spaces associated to representations of $...

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### Principal bundles and fibre bundles

Let $\pi_P:P\rightarrow M$ a principal $G$ (right action) bundle. Let $F$ be a manifold with a left action of $G$. Then we have the notion of associated fibre bundle over $M$ whose fibre is $F$. I do ...

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### Chern-Weil theory and Weil homomorphism of principal bundle

In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $G$ bundle. In this book, they use connection on a principal bundle to ...

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### Classifying map of a simple circle bundle

Let $\mathbb{K}_0 \subset \mathbb{K}$ be two tori (subtori of $(S^1)^n$). We suppose that $\mathbb{K}_0$ is obtained from $\mathbb{K}$ by the following procedure: consider, on the lie algebra $\text{...

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### Associated bundle construction and classifying space

Let $\theta:G\rightarrow H$ be a morphism of Lie groups.
Given $G$ we have classifying space $BG$ and given $H$ we have classifying space $BH$. This $\theta:G\rightarrow H$ gives a map $B\theta:BG\...

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### Torsor descriptions of $Bun_G$

The moduli stack $Bun_{SL_n}X$ of $SL_n$-principal bundles over a projective curve $X$ is a $\mathbb{G}_m$-torsor over the sublocus of $Bun_{GL_n}X$ corresponding to vector bundles with zero ...

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### Morphism of Lie groups $\theta:G\rightarrow H$ giving an equivalence of categories $BG\rightarrow BH$?

Given a morphism of Lie groups $ \theta:G\rightarrow H$ and a principal $G$ bundle $ \pi:P\rightarrow M$ there are (at least) two ways to assign a principal $ H$ bundle.
See that the morphism of Lie ...

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### Different definitions of a structure on principal bundles

Let $P\to B$ be a principal $G$-bundle and $\psi:H\to G$ a homomorphism of topological groups. A $\psi$-structure for $P$ can be defined in two different ways. I am trying to prove their equivalence.
...