Questions tagged [fibre-bundles]

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

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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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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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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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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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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 ...
annie marie cœur's user avatar
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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^...
annie marie cœur's user avatar
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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 ...
Elizeu França's user avatar
3 votes
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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^...
wonderich's user avatar
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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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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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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 ...
R Mary's user avatar
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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 ...
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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 := \...
phydev's user avatar
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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 ...
Yuhang Liu's user avatar
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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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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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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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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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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):...
Mohammad Ghomi's user avatar
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196 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;\...
ort96's user avatar
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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 ...
C. Zhihao's user avatar
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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 ...
Max Reinhold Jahnke's user avatar
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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 ...
Ali Taghavi's user avatar
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1 answer
385 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 ...
ort96's user avatar
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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 ...
ort96's user avatar
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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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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 ...
Krishna's user avatar
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12 votes
2 answers
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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 ...
ychemama's user avatar
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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 ...
ychemama's user avatar
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1 answer
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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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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}:...
54321user's user avatar
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3 votes
1 answer
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Disk bundles over surfaces - extending automorphisms of the boundary over the whole space

Let $\Sigma$ be a closed connected orientable surface of genus $g \geq 2$. I have been told that every diffeomorphism $\phi: \Sigma \times S^1 \to \Sigma \times S^1$ extends to a diffeomorphism $\Phi ...
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introduction textbook to the Laplacian on a circle bundle

I am looking for an introduction to spectral theory of $\Delta$ on a circle bundle over a compact M. Is there an analog of Selberg trace formula?
alexander's user avatar
8 votes
2 answers
596 views

In which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?

In what follows, all manifolds are smooth, Hausdorff, paracompact, connected and oriented, and all maps between any two of them are assumed to be smooth. Let $\pi:E\rightarrow M$ be a fiber bundle ...
Pedro Lauridsen Ribeiro's user avatar
5 votes
2 answers
1k views

Triviality of the adjoint and endomorphism bundles

Let $P$ be be a principal bundle over a manifold $M$ with structure group $G$, where $G$ is a Lie group. Let $E = P\times_{\rho} \mathbb{R}^{k}$ be a vector bundle associated to $P$ through a faithful ...
Bilateral's user avatar
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6 votes
1 answer
250 views

Simplicial approximation of a fibration

I am interested in the simplicial approximation of Serre or Hurewicz fibrations (or even fibre bundles). Let's assume $E$ and $B$ are finite simplicial complexes (or their associated geometric ...
D1811994's user avatar
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4 votes
1 answer
645 views

Lifting cellular structures to fibrations, fibre bundles or coverings

It is a well known result in Algebraic Topology that given a covering space $E\to B$ where the base has a CW-structure, then the total space can be given a CW-structure (see for example Theorem 8.10 ...
D1811994's user avatar
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Deformation of projective bundles

Let $\pi:\mathcal{X} \to \mathbb{P}^1$ be a flat, projective family of noetherian schemes with generic fiber a smooth, projective variety. Let $p:\mathcal{Y} \to \mathcal{X}$ be another flat, ...
Ron's user avatar
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4 votes
1 answer
322 views

Surface bundles over surfaces with(out) flat structure

I vaguely remember that I once attended a seminar or conference talk in which it was mentioned that the following question is open. Is there a (smooth) surface bundle over a surface $\Sigma_h \to E \...
Jens Reinhold's user avatar
2 votes
0 answers
237 views

Special orthogonal groups over spheres

In Norman Steenrod's book "The Topology of Fibre Bundles", on page 37, one can find the following conjecture: if $n$ is a power of two then the fibre bundle with the projection $SO(n)\to SO(n)/SO(n-1)=...
William of Baskerville's user avatar
4 votes
1 answer
388 views

Second differential in long exact sequence for Cech cohomology for nonabelian groups

I do not really think it fits MO, but I posted it in MathStackExchange with little success, so... Assume that we have a short exact sequence of, say, Lie groups $1\rightarrow A\rightarrow B\...
berndt's user avatar
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4 votes
0 answers
111 views

Bundle structures on spheres

Given a positive integer $n$, there is a well known free action of $\mathbb T^1$ on $\mathbb S^{2n-1}$ due to Hopf, which makes $\mathbb S^{2n-1}$ a fibre bundle with the fibre $\mathbb T^1$. Moreover,...
William of Baskerville's user avatar
7 votes
1 answer
1k views

Morphisms of principal bundles with different structure groups and associated bundles

Consider a pair of principal bundles $P \to M$ and $P' \to M'$ with groups $G$ and $G'$, respectively. A morphism from $P$ to $P'$ is a pair $(\Phi, \phi)$ where $\phi: G \to G'$ is a Lie group ...
ಠ_ಠ's user avatar
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4 votes
1 answer
193 views

Exhausting fibrations by fibrations with finite type fibers

If $E \to B$ is a fibration, is it always possible to find fibrations $E_n \to B$ with maps over $B$ $E_n \to E_{n+1}$ such that $E$ is weakly equivalent to the homotopy colimit of the $E_n$'s and ...
2223's user avatar
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3 votes
1 answer
652 views

Vector Bundle Structure

A smooth vector bundle of rank $n$ is usually defined as a smooth map $p: E \longrightarrow B$ together with a real vector space structure on each fiber $E_b := p^{-1}(b)$ such that: (Local ...
paeolo's user avatar
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4 votes
1 answer
595 views

Does every disc bundle come from a vector bundle?

Kosinski in his book "Differential Manifolds" states: "A closed tubular neighbourhood $E$ of a compact submanifold $M$, which is closed neighbourhood in $N$, can always bee realised as a closed disc ...
Igor Ernst's user avatar
2 votes
0 answers
204 views

Explicit description for dual to cohomology class

Here's an inductive definition of a particular Bott tower manifold from https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/on-a-construction-in-bordism-...
Grigory Solomadin's user avatar
3 votes
0 answers
450 views

Cohomology rings of 3-torus bundle

The 3-torus bundle $E$ is a fiber bundle with 3-torus $T^3=S^1\times S^1\times S^1$ as the fiber, and $S^1$ as the base space. Then what are the cohomology rings of $E$: $H^*(E;\mathbb{Z}_n)=?$ and $H^...
Xiao-Gang Wen's user avatar
4 votes
1 answer
326 views

Some dynamical and Bundle questions arising from certain map $P:TS^{n}\to S^{n}$

Define the map $$P:TS^{n}\to S^{n} \;\;\;\text{by}\;\; P((x,v))=\frac{x+v}{\parallel x+v \parallel}$$ where $$TS^{n}=\{(x,v)\in S^{n} \ \times \mathbb{R}^{n+1}\mid v \perp x \}$$ This map is ...
Ali Taghavi's user avatar
5 votes
2 answers
504 views

A $\mathbb{R}^{n}$ -fiber bundle which do not admit a n-dimensional vector bundle structure

Is there a fiber bundle $(E,B, \mathbb{R}^{n})$, with typical fiber $\mathbb{R}^{n}$, such that there is no any $n$-dimensional vector bundle structure on the pair $(E,B)$? That is there ...
Ali Taghavi's user avatar

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