The principal-bundles tag has no usage guidance.

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### Vector bundle with a perfect pairing and ($\mathbb Z/2$, $SL_r$)-bundle

I think this is a well knowing result but I can't find any reference,
Let $(E,q)$ be a vector bundle with a non degenerated quadratic form $q:E\rightarrow E^*$ with trivial determinant, suppose ...

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135 views

### Detecting torsion-classified bundles by differential invariants

The following is based on a loose understanding of the nuts and bolts that go into Chern-Simons theory, so bear with any vagueness on my part.
Suppose I have a principal $G$-bundle $P\to M$ and I ...

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**2**answers

377 views

### Generalising the Penrose Twistor Fibration

As is well known, there exists a fibration $\mathbb{CP}^3 \to S^4$, of the four sphere by complex projective $3$-space, called the Penrose twistor fibration. Does this fibration admit a "canonical" ...

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**1**answer

169 views

### Automorphism group of a fiber bundle surjects onto diffeomorphism group?

This should surely be well-known by I have not been able to find a good reference to the following question: Given a smooth fiber bundle $\pi\colon P \longrightarrow M$ over a smooth manifold $M$ with ...

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98 views

### Certain principal bundle structure on $\mathbb{R}^{n} \setminus \{0\}$

I ask this question in MSE and I received no answer, so I repeat it here:
Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a ...

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**2**answers

133 views

### What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?

Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.
Is it true that if $A$ is the connection 1-form of ...

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**2**answers

185 views

### Lindel's theorem for semisimple simply connected G

Let $k$ be a field.
$G/k$ be a simply connected semisimple algebraic group.
Let $X/k$ be a smooth affine $k$-scheme.
Question: Is every principal $G$ bundle on $X\times {\mathbb A}^1$ a pull back ...

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152 views

### Principal bundles and Subriemannian Geometry

In sub-Riemannian geometry, one considers manifolds $P$ equipped with a subbundle $\mathcal{H}$ of $TP$, the horizontal distribution. One then has a Riemannian metric only on this distribution ...

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81 views

### Twisting stable maps to C* equivariant space by a line bundle

Let $X$ be a $\mathbb{C}^*$-equivariant algebraic variety. Then there is a notion of a map to $X$ twisted by a line bundle. Namely, let $B$ be a variety and $L/B$ a line bundle. Let $P_L=L\setminus ...

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**1**answer

169 views

### Metric, torsion free connections on principal bundles

I hope this is not too elementary, but I have asked this question at the math.stack site, but I have obtained no answers.
Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold ...

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119 views

### When does an algebraic space that is a torsor over a scheme have to be a scheme?

In Group actions on stacks and applications (Section 4 of part A), M.Romagny gives a definition of $G$-torsor over a scheme $S$ in which the total space need not be a scheme, just an algebraic space. ...

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**1**answer

200 views

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

I asked this question on math.stackexchange a week ago, but did not get an answer.
First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general ...

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80 views

### Cohomology of a flat principal connection

Let $M$ be a compact manifold, $G$ a compact Lie group, $P\to M$ a principal $G$-bundle and $A$ a flat principal connection on $P$. Then $(\Omega^\bullet(M;\operatorname{ad}P),d_A)$ forms a cochain ...

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101 views

### Identifying a cohomology class arising from a Postnikov decomposition of BU(2)

For various reasons I'm currently interested in Principal $U(n)$-bundles $U(n)\hookrightarrow E\xrightarrow{p} S^m\times X_g$ over the Cartesian product of an $m$-sphere and a closed oriented surface ...

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205 views

### The unit tangent bundle of 2- or 4-manifolds as a principal $S^{1}$- or $S^{3}$-bundle

What type of obstructions have been studied so that the unit tangent bundle of a Riemannian 2-(4-)manifold have a structure of a principal $S^{1}$-($S^{3}$-)bundle?

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363 views

### Clutching functions and Classifying maps

Let $E\xrightarrow{p} \Sigma X$ be a principal G-bundle over a suspension. Write $\Sigma X= C_+X\cup_X C_-X$. Then there are trivialisations of the restrictions $E|_{C_+X}\cong C_+X\times G$, ...

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190 views

### Universal bundles: construction of the map associated to a group homomorphism

For a Lie group $G$ let $EG \to BG$ denote the universal bundle. A Lie group homomorphism $\rho: G \to H$ determines a map $B \rho: BG \to BH$ as the classifying map for the principal $H$-bundle $EG ...

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151 views

### Differentiable structure on the Gauge group of a principal bundle?

I am currently reading this paper in which we have a map $g:I\rightarrow Gauge(P)$ for some principal bundle $P$ which is differentiated. I am looking for a reference or explanation what the "most ...

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141 views

### $v_1$-periodic homotopy and principal bundle classification

This question came back to my mind while pondering this MO question. The classification of principal bundles is seriously difficult because of our lack of understanding of homotopy groups of compact ...

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**1**answer

156 views

### Associating a principal bundle to a torsor

in arXiv:math/0212266, Moerdijk defines a torsor to be a sheaf $\mathcal{S}$ on $X$ with a freely transitive left-action of a sheaf of groups $\mathcal{G}$, such that $X =\bigcup \{ U \in ...

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92 views

### Proof of Lemma in “Harmonic maps and the self-duality equations” by Donaldson

I am referring to this paper by S. K. Donaldson. I could not find a freely available version, hence I feel uncomfortable to copy & paste parts of his paper, and won't do so. Nevertheless, I will ...

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**1**answer

93 views

### Decomposing connections on extensions of the frame bundle

I have posted this question on math.stackexchange, without success. I'll make it brief:
Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, ...

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87 views

### Euler number of the complex of basic forms

Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ...

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267 views

### What are global sections of the determinant bundle on the Beilinson-Drinfeld Grassmannian?

Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$, and let $G$ be a reductive group over $\mathbb{C}$. Let $Gr_{X,n}$ be the Beilinson-Drinfeld Grassmannian (for n points in $X$), which ...

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**1**answer

83 views

### Normalizing the value of a principal connection at a point

Let $\nabla$ be a symmetric, linear connection on a smooth manifold $X$.
If $p \in X$ is any point, on a normal chart for $\nabla$ around $p$ it holds:
$$ \Gamma_{ij}^k (p) = 0 \ , $$ where ...

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**2**answers

241 views

### Classifying map of a principal bundle

Given a topological group, say $G$, it is well known that the classifying space $BG$ classifies $G$ principal bundles. Under some mild assumption on $G$, one model of $BG$ is the geometric realization ...

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249 views

### Are there principal $G$-bundles whose holonomy group is $G$?

While studying universal constructions on principal bundles, I've stuck on a quite a basic question, namely:
Given a Lie group $G$, does there exist a principal $G$-bundle $\pi \colon P \to B$, for ...

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336 views

### Example: Principal G bundle that is not Zariski locally trivial, G not finite and G simply connected

Let $G$ be an affine algebraic group over $\mathbb{C}$. It is well known that when working with principal $G$ bundles it is too restrictive to require bundles to be locally trivial in the Zariski ...

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231 views

### Principal bundles and Čech cohomology with non-good open covers

I'm trying to compute characteristic classes of principal bundles by defining transition functions and computing them in Čech cohomology. However, it seems all the constructions are defined in terms ...

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138 views

### Obstruction to this gauge choice of the connection of a vector bundle

Let $M$ be a compact manifold with a nowhere-vanishing vector field $R$. Consider principal $G$-bundle $P$ over $M$, and $\mathcal{A}$ being the space of irreducible connections.
Let me denote a ...

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147 views

### Split real form of $SL(2,\mathbb{C})$ description of the two sphere?

If we denote the parabolic subgroup of $SL(2,\mathbb{C})$ by $P$, then we have the well known isomorphism $SL(2,\mathbb{C})/P \simeq S^2$, where $S^2$ is the two sphere. Now the compact real form of ...

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315 views

### Representation variety vs. space of flat connections

The holonomy provides a bijection from
the space of flat G-connections (modulo gauge equivalence) on a trivial G-bundle over M
to
a connected component of the representation variety ...

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**1**answer

100 views

### Actions of compact Lie groups on (possibly but hopefully not) non-regular spaces

Suppose $G$ is a compact Lie group acting freely on a topological space $Q$ (about whose separation conditions I make no assumptions) and the qoutient $Q/G$ is known to be completely regular Hausdorff ...

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590 views

### What does the moduli stack of G-torsors over the multiplicative group look like?

I am an algebraic topologist and am trying to understand some computations related to p-adic complex K-theory and equivariant K-theory. However this has led me into the world of algebraic geometry ...

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455 views

### Spaces of symplectic embeddings: Bundle? Smoothness?

Let $(M, \omega)$ and $(N, \sigma)$ be two symplectic manifolds, $M$ compact and without boundary. Consider the space $$ \mathcal{E} = \mathrm{Emb}((M, \omega), (N, \sigma)) $$ of all smooth ...

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235 views

### Formula for the curvature of an induced connection

Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles ...

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481 views

### How many principal bundles are there over a given base?

I'm currently considering principal bundles and classifying spaces in the context of gauge theory and a crucial question came to my mind:
Is there a way to say how many (isomorphism classes of) ...

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**1**answer

86 views

### Non-compact structure group and compactly supported gauge transformations

Let $\pi\colon P\to X$ be a locally trivial principal $G$-bundle over a Hausdorff paracompact space $X$, where $G$ is a topological group (we work in the category of topological spaces, as I do not ...

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311 views

### Are principal bundles isotrivial?

Let $U$ be a $k$-scheme, where $k$ is a field. Let $G$ be a smooth affine $k$-group. Recall that a principal $G$-bundle over $U$ is a smooth surjective $U$-scheme $E$ with an action of $G$ on $E$ such ...

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253 views

### The relation between the heat kernel on the principal bundle and the heat kernel on the base manifold

This question is mainly about Section 5.2 of the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne.
Let $M$ be a compact Riemannian manifold without boundary and $P\rightarrow M$ ...

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226 views

### State of the game : cohomology of principal bundles

I would like to know what has been done in terms of specific calculations for the cohomology of principal bundle. For instance, it is known (see e.g. Greub, Halperin and Vanstone's "Connection, ...

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181 views

### How to characterize this particular kind of bundle?

I am considering the following situation.
Let $M_5$ be a 5-dimensional manifold which is an $S^1$ principal bundle over 4-manifold $M_4$. For instance, $M_5 = S^5$ and $M_4 = \mathbb{CP}^2$ with ...

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412 views

### What is “Data” involved in a mathematical construction?

What exactly do mathematicians mean when they refer to "the data" involved in a construction?
I've encountered this many times and I can usually figure out what's going on, but I am curious about the ...

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211 views

### Confusions over the definitions of universal bundle and characteristic class

In most references, only a principal G-bundle is called universal (ie every other bundle can be pullbacked from this one, or an equivalent definition).
Does it make sense to speak of a universal F-G ...

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539 views

### Trivializing principal bundles on a curve over a finite field

This is related to my question Adelic description of moduli of $G$-bundles on a curve.
Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and $G$ a ...

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189 views

### Twisting an object P by an H-Torsor I

I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization.
The Statement
Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a local homeomorphism ...

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325 views

### What are the symmetries of a principal homogeneous bundle?

Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where:
$G$ is a Lie group, and $H$ ...

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256 views

### Numerable covers from the point of view of Grothendieck topologies

Let $G$ be a topological group. Recall that its classifying space $BG$ is a CW-complex which is the base of a locally trivial principal bundle of group $G$, with contractible total space $EG$. It ...

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369 views

### Higgs bundle and stable bundle

Let $(E,\phi)$ be a $G$-Higgs bundle $\phi\in H^{0}(X,ad(E)\otimes D)$ where $D$ is a divisor on X.
I suppose that $(E,\phi)\in \mathcal{M}^{ani}$ the anisotropic locus.
In particuler, this bundle ...

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358 views

### Connection Transformation Formula; Degree 3 Cech Cohomology

While reading through Brylinski, as in all of my posts, I am trying to understand the following equation:
$ g_* \tilde{\theta} = \tilde{\theta} - g^{-1} dg$
Setting
I have a principal ...