# Questions tagged [fibre-bundles]

for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.

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### Classifying space of semidirect product of groups

Assume that $G$ and $H$ are two groups and $G\rtimes _\phi H$ is their semidirect product. My question is, how does the classifying space $B(G\rtimes_\phi H)$ of $G\rtimes _\phi H$ relate to $BG$ and $...

**3**

votes

**0**answers

70 views

### Torus bundle over spheres

I was wondering what is the classification of all torus bundles over spheres? That is, to classify the fibration
$$
T^m \hookrightarrow M \to S^n.
$$
It is well known that if $n=1$, all fibrations ...

**2**

votes

**0**answers

99 views

### Fiber-bundle : continuity of transition maps and inverse in general

Let $(E,\pi,B)$ be a locally trivial fibration, with fiber a topological space $F$, $\Phi_i$ and $\Phi_j$ two trivializations over $U_i$ and $U_j$. The transition map from $i$ to $j$ is the ...

**4**

votes

**1**answer

150 views

### Smooth structure on the space of sections of a fiber bundle and gauge group

Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with ...

**4**

votes

**1**answer

138 views

### The existence of the extension of a non-trivial line bundle

In Three Dimensional Gravity Revisted, Witten studied the Abelian Chern-Simons theory in three dimensions.
Let $W$ be a three dimensional manifold. Let $\mathcal{L}$ be a non-trivial line-bundle over ...

**2**

votes

**0**answers

31 views

### Smoothings over a real interval

I am asking if somebody knows if the following kind of objects are studied somewhere or if there is some kind of obvious obstruction for them to appear.
Let $(X,0) \subset \mathbb{C}^n$ be a germ of ...

**5**

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87 views

### Induced new structures on Poincare dual manifolds

"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows
Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...

**5**

votes

**2**answers

119 views

### Lifting a diffeomorphism into a spinor bundle automorphism

I know several papers that treat this, but it seems that most of these papers do things very differently with quite different conclusions, so I am confused.
Basically, when one tries to do classical ...

**1**

vote

**0**answers

66 views

### Explicit description of the scheme obtained by relative gluing data over a base scheme

I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...

**3**

votes

**1**answer

97 views

### Naturality of minimal model of a fibre bundle

$\require{AMScd}$
For rational fibrations $F \rightarrow E \rightarrow B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's
$$
\begin{CD}
...

**0**

votes

**1**answer

84 views

### Is $TS^2\setminus Z$ a $S^2$- fibre bundle on the puntured plane?(Swapping the role of fibre points and base space)

Let $X=TS^2\setminus Z$ where $Z$ is the zero section of the tangent bundle of $S^2$.
Is there a $S^2$- fiber bundle structure on $(X,\mathbb{R}^2\setminus\{0\},q)$ for some continuous fibre map $q$?

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votes

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189 views

### When is the Chern class of a principal $G$-bundle same to the Chern class of the associated complex vector bundle?

Let $P\rightarrow M$ be a principal bundle, with structure group $G$. If $G$ has a representation $\rho:G\rightarrow GL(n,\mathbb{C})$, then we can define its associated vector bundle $E=P\times_{\rho}...

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votes

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74 views

### Representation on square integrable sections of a principal bundle

Let $X\rightarrow Y$ be a smooth principal $G$-bundle for some Lie group $G$. Then $L^2(X)$ has a natural $G$-action determined by fibrewise action of $G$ on $X$.
We have an abstract isomorphism of ...

**2**

votes

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58 views

### Extension of a given section and obstruction cocyles

Let $p:E \to X$ be a fibration with the fiber $F$ where $X$ is a CW-complex. Denote by $U$ the set $U:=D^n \times \{0\} \cup S^{n-1} \times I$ (part of a cylinder) and let $\tilde{f}:U \to E$ be a ...

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75 views

### Lie algebra bundle associated to a Lie group bundle

I was reading something(page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.
I am not comfortable with these notions and google gave http://www.pphmj.com/Images/...

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votes

**1**answer

110 views

### Proper locally trivial bundle is injective on Chow groups

If $X\to Y$ is a map of varieties that is Zariski-locally isomorphic to a projection $U\times P\to U$ with $P$ (smooth) proper, I think the pullback $A_{\bullet}(Y)\to A_{\bullet}(X)$ is supposed to ...

**3**

votes

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208 views

### Is a Difference of Fiber Bundles a Fiber Bundle?

I have a seemingly very basic question in differential topology, but I could not find the answer by a short google search.
Let $M,N$ be smooth manifolds, and let $f:M\to N$ be a smooth fiber-bundle,
...

**4**

votes

**1**answer

77 views

### Homological class of the fiber in the total space of the one circle bundle

It is a well known fact that (isomorphism classes of) princial $\mathbb{S
}^1$-bundles over a base space $B$ are classified by $B$'s second integral cohomology, $H^2(B;\mathbb{Z}$), by the Euler class....

**4**

votes

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120 views

### Instanton configurations of self-dual and anti-self-dual instantons interplay

Yang-Mills gauge theory is given by the action
$$ S_\text{YM}[A] = \int_M\mathrm{Tr}_\mathfrak{g}(F\wedge \star F)$$
whose Euler-Lagrange equations are the classical equations of motion. The classical ...

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vote

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100 views

### Conventions / Normalizations of Yang-Mills Field Theories

Let the spacetime be 4-dimensional.
In the usual Maxwell theory of Abelian gauge fields $A$, where field strength $F=dA$ one considers the Maxwell action written as
$$
S_{Maxwell}\equiv\int
-\frac{...

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votes

**0**answers

127 views

### Reference request: classification of principal bundles

I found this theorem in the appendix of Friedrich's "Dirac Operators in Riemannian Geometry"
(Second homotopy classification theorem)
1. For every topological group $G$ there exists a universal ...

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votes

**0**answers

52 views

### Fibre restriction and Leray–Hirsch

Let $\mathbb{C}P^h\to E\to B$ be a bundle which is cohomologically trivial via Leray–Hirsch, i.e. if we denote $H^*(\mathbb{C}P^h)\cong \mathbb{Z}[\mu]/\mu^{h+1}$, we have a class $\chi\in H^2(E)$ and ...

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vote

**1**answer

94 views

### Extend a bundle “trivially”

Suppose I have a fibre bundle $E\to B$ with compact fibre. Furthermore, $B$ is open in a larger, compact space, e. g. $B\subseteq B'$. I want to get a map $E'\to B'$ (not a bundle any more!) with
$E'|...

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115 views

### Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(2)$ or $BO(2)$

Thanks to a suggestion by @Igor Belegradek, I am interested also in a simpler problem of this earlier question 301523, by knowing what can we say about the classification of fibrations for classifying ...

**5**

votes

**1**answer

265 views

### Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(3)$ or $BO(3)$

I am interested in knowing what can we say about the classification of fibrations for classifying spaces $B^2M \equiv B^2\mathbb{Z}_2$ and $BG \equiv BSO(3)$ or $BO(3)$.
Here we can take either:
$B^...

**2**

votes

**1**answer

214 views

### Does every fiber bundle admits flat bundle structure?

It is well known that a bundle which admits a compatible foliation* supports a flat connection and the converse is also true (every smooth and finite dimensional). The question is:
Does every bundle ...

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votes

**2**answers

174 views

### Signature of the manifold of the multiple fibrations over spheres

We can define the signature of a manifold in $4k$ dimensions.
1) If I understand correctly, the signature $\sigma$ of the manifold of the product space of spheres would always be zero:
$$\sigma(S^...

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**0**answers

79 views

### Transverse $S^1$ actions on mapping tori

Up until now I have thought that the existence of a transverse $\mathbb{S}^1$ action on a symplectic mapping torus implies that the mapping torus is trivial. Unfortunately I also came up with a ...

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190 views

### Models for Eilenberg-MacLane space K(Z,3)

Denote by $U(H)$ and $PU(H)$ the unitary and projective unitary groups on an infinite-dimensional Hilbert space $H$. Recall that $U(H)$ is contractible by Kuiper's theorem and that $PU(H)$ is a $K(Z,2)...

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vote

**1**answer

149 views

### The holonomy map associated to a mapping torus

So I have a rather embarrassing problem, which is not really a "problem", so much as a mental block I seem to be unable to overcome. I am trying to understand the "holonomy map" of a mapping torus. To ...

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194 views

### Why is any $G$-resolution a principal $G$-bundle?

In the article The Cohomology of Classifying Spaces of H-Spaces by M. Rothenberg and N. Steenrod (https://projecteuclid.org/euclid.bams/1183527356) it is stated as a theorem that if $G$ is a ...

**6**

votes

**1**answer

229 views

### Action of the spin covariant derivative on gamma matrices?

How does the spin covariant derivative $\nabla^S_{\mu}$ act on gamma matrices satisfying: $\{\gamma^{\mu},\gamma^{\nu}\} = g^{\mu\nu}$, i.e.
$$\nabla^S_{\mu}\gamma^{\nu} = ?$$
where $\nabla^S := \...

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178 views

### Recovering SU(2)-space when the orbit space is a 3 sphere with 3 singular orbits

Background: Consider $SU(2)$ action on the 6-dim flag manifold $M=SU(3)/T^2$ via left multiplication. We view $SU(2)$ as a subgroup of $SU(3)$ corresponding to $2\times 2$-block. The action is just ...

**4**

votes

**1**answer

246 views

### A map from a symmetric product of a curve to its Jacobian

Let $C$ be a smooth projective curve over an algebraically closed field $k$, of genus $g$.
It is well known that, after fixing a point $p_0$, the map $C^{(n)}\to J$ sending $\{a_1,\dots,a_n\}$ to $[...

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119 views

### Projective unitary flat structures of $\mathbb{P}^1$-bundles on Riemann surfaces

Narasimhan and Seshadri proved a rather surprising result about vector bundles on a compact connected complex manifold $X$. That is
Two holomorphic vector bundles arising from unitary representations ...

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349 views

### Hartshorne's Conjectures about Algebraic Bundles?

In 1978 Hartshorne published a list of 26 open problems about algebraic bundles on projective spaces [Hart], proceeding from an Oxford conference organized by Atiyah.
I understand that many of these ...

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62 views

### Two questions regarding flat fibre bundles and the corresponding group action on the fibre

Let $F$, $B$ be smooth, closed manifolds and $\phi:\pi_1(B) \rightarrow Aut(F)$ a smooth group action of the fundamental group of $B$ on $F$.
Consider the flat fibre bundle $E_\phi := \widetilde{B} \...

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votes

**1**answer

378 views

### Equivariant sections of fiber bundles

One of the fundamental facts in fiber bundle theory is the following result for existence and extension of sections (see Thm. 9 in this paper of Palais, and compare with Thm. 12.2 in Steenrod's book):...

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**1**answer

160 views

### Cup product on flat fiber bundles vs cup product on the corresponding Serre spectral sequence

Let $F \rightarrow E \rightarrow B$ be a flat fiber bundle, $E, F, B$ closed manifolds. Consider $H^*(E, \mathbb{Q})$ and the corresponding Serre spectral sequence with isomorphism $$(*) \ \ \ H^n(E;\...

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votes

**1**answer

110 views

### non-simple local coefficient system on a fibration of classifying spaces

Long story short; I posted in MSE
https://math.stackexchange.com/questions/2500745/local-system-of-coefficients-on-a-fibration-of-classyfing-spaces
It is well known that if $G$ is a lie group ...

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votes

**0**answers

157 views

### Smooth trivialization of smooth Hilbert bundles

In page 67 of Topology and Analysis by Booss and Bleecker, it is claimed that any Hilbert bundle is topologically trivial. Clearly, any smooth Hilbert bundle over a smooth manifold is topologically ...

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216 views

### Obstructions for existence of a fiber wise covering space structure( A bundle of covering spaces)

Let $S^n \times \mathbb{T}^n$ be the trivial Torus bundle over $S^n$.
Assume that we have a continuous fiber preserving map $\phi :TS^n \to S^n \times \mathbb{T}^n$ which restriction to each fiber ...

**2**

votes

**1**answer

245 views

### Compute cohomology of flat fiber bundles - does this always work?

Edit: This has been answered in this thread Are there compact flat fiber bundles with "truly" infinite structure group?.
Setting
Let $p: E \rightarrow B$ be a flat fiber bundle with fiber ...

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119 views

### Equivalence of Flat Fiber Bundles vs Equivalence of Group Actions on the Fiber

Let's consider all flat fiber bundles with base space $B$ and fiber $F$, where $B$ and $F$ are compact and at least CW-complexes. (perhaps even topological/smooth manifolds if that helps)
All those ...

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183 views

### Complete proof of Homotopy invariance of a numerable fiber bundle based on CHP

Homotopy invariance of numerable fiber bundle: let $\xi = (E,π,B)$ (i.e. $\pi : E \to B$) a numerable topological fiber bundle, $X$ a topological space, and $f,\,g : X \to B$ two continuous ...

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votes

**1**answer

162 views

### Extensions of Fried's Theorem: Surface bundles over Circles and Flows on 3 Manifolds

The Thurston Polytope provides a way to organize information about the embedded surfaces living in a 3-manifold.
His amazing theorem, often called the "Fibered Faces" Theorem, says that if you have ...

**8**

votes

**2**answers

533 views

### Is a topological fiber-bundle, whose total space admits a retraction onto a fiber, trivial?

Let $\xi = \pi \colon E \to B$ a topological fiber bundle with connected base $B$, $E_x = \pi^{-1}(x)$ the fiber at $x \in B$, $j \colon E_x \hookrightarrow E$ the canonical injection, and let suppose ...

**7**

votes

**1**answer

318 views

### Are topological fiber bundles on the same base with homeomorphic fibers isomorphic?

First I apologize because this is not a research question, but I can't get any answer on MathStackExchange...
Let $\pi \colon E \to B$ and $\pi' \colon E' \to B$ two topological fiber bundles on the ...

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votes

**1**answer

115 views

### Embeddings of fiber bundles

While reading GMTW - The homotopy type of the cobordism category (https://arxiv.org/abs/math/0605249, p. 16) I found the following passage:
Lef $f:W\to \mathbb{R}$ be the projection. Then $(\pi_2,...

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172 views

### How can I find the differential in the Serre spectral sequence for this sphere fibration?

Consider the assocaited sphere bundle $$S(E) \to \mathbb{P}^n$$ for the vector bundle $\mathcal{O}(k)\oplus \mathcal{O}(l) \to \mathbb{P}^n$. Is there a way to determine the differentials
$$
d_4^{p,m}:...