Questions tagged [fibre-bundles]
for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.
336
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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 ...
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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^...
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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 ...
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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^...
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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 ...
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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 := \...
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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 ...
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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):...
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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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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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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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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 ...
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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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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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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 ...
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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 ...
7
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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 ...
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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}:...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 \...
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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)=...
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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\...
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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,...
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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 ...
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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 ...
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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 ...
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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 ...
2
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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-...
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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^...
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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 ...
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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 ...