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Questions tagged [fiber-bundles]

For questions about fiber bundle, which is a space that is locally a product space, but globally may have a different topological structure.

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Show that a sphere bundle over S^2 is trivial

I want to show that the following sphere bundle is trivial. Consider $S^2=\{(z,x)\in \mathbb C\times \mathbb R \mid |z|^2+x^2=1\}$ with the open cover ${U_1,U_2}$ of $S^2$, where $U_1=S^2\setminus (0,...
JerryCastilla's user avatar
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Local cross-sections of a Hopf bundle

The group $\mathrm{Spin}(9)$ is the total space of the fibre bundle over the sphere $S^{15}$ with fibres that are copies of $\mathrm{Spin(7)}$. It has no continuous global cross-sections. Is it ...
Anatoliy Malyarenko's user avatar
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Total space of the frame bundle always an almost complex manifold in $dim >2$?

I seem to remember reading that the total space of the frame bundle of a smooth Riemannian manifold always admits an almost complex structure in dimensions greater than 2. However now I can't seem to ...
R. Rankin's user avatar
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2 answers
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Is the change of trivialisations on a principal $G$-bundle given above each basepoint by a right-multiplication with a group element?

Let $\pi: M \rightarrow B$ be a principal $G$-bundle. Suppose $(U_i, \phi_i)$, $(U_j, \phi_j)$ are two (equivariant) trivialisations $\phi_i: \pi^{-1}(U_i) \rightarrow U_i \times G$ (similarly for $\...
rosecabbage's user avatar
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Equivalence between two definitions of a Seifert Fibered Homology Sphere

I am reading Savaliev's Invariants for Homology 3-spheres. Here, he defines the Seifert Fibered Homolgy Sphere as $\Sigma(a_1,...,a_n) = V_B(a_1,...,a_n) \cap S^{2n-1}$ where $V_B(a_1,...,a_n) = \{b_{...
user13121312's user avatar
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Equivalent definition for bundle orientability.

We know there are several equivalent definition for orientability of manifolds(see Lee's introduction to smooth manifolds): Existence of a choice of orientation at each point $p\in M$, such that ...
Eric Ley's user avatar
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Decomposition of Differential Forms

Bott-Tu's Differential forms in algebraic topology says every differential form over $\mathbb R^n\times \mathbb R=\{(x,t)\}$ uniquely decomposed into two types of forms, one type with $d t$ and one ...
Eric Ley's user avatar
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4 votes
1 answer
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Definition of Reduction of a Structure Group Implies Triviality?

Let $\pi:E\rightarrow B$ be a (smooth) principle bundle with structure group $G$. By definition, there exists a reduction of the structure group to a subgroup $H<G$, if there exists a global ...
LarsB's user avatar
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Group $U(1)$ as a principal bundle whose base is $U(1)$ and the fiber is ${\mathbb Z}_2$

Why is the principal bundle with $U(1)$ as its base and ${\mathbb Z}_2$ as a fiber still isomorphic to $U(1)\,$? Stated alternatively, why is it that $U(1) = U(1)/{\mathbb Z}_2\,$? I would greatly ...
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Is there a simple description of the total space of a principal S^1 bundle over a compact surface?

It is known that principal $S^1$-bundles over a compact surface $\Sigma_g$ are classified by their Chern classes in $H^2(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}$. When the Chern number is zero, the ...
Rei Henigman's user avatar
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Understanding the rolling motion of a circle on $\mathbb{R}$ in terms of parallel transport and Ehresmann connections

I am trying to understand a toy example to help build my intuition about connections on fiber bundles and parallel transport. My main issue is trying to understand if and how the "no-slip" ...
Tob Ernack's user avatar
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2 votes
2 answers
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Circle bundle over $\Bbb CP^n$ whose total space is $S^{2n+1}$

Viewing $S^{2n+1}$ as the unit sphere in $\Bbb C^{n+1}$, the natural projection $S^{2n+1}\to \Bbb CP^n$, the generalized Hopf fibration, is a circle bundle. What I am curious about is its converse. ...
user302934's user avatar
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Is the torus as a bundle over the circle a vector bundle?

I was looking for an example of a smooth fiber bundle over a manifold that is not a vector bundle. My idea was that $S^1 \times S^1 \overset{\pi}{\rightarrow} S^1$ has $S^1$ as Fibre, which doesn't ...
Pastudent's user avatar
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Metric of Normal Bundle

If I had a circle bundle over some base manifold $\mathcal M$, I could write the line element in local coordinates as $$ds^2=g_{ij} dx^i dx^j+\phi(x)(dz^2+A_i dx^i)^2,$$ where $g_{ij}$ is a Riemannian ...
arow257's user avatar
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Prove that in Hopf bundle $p:S^3\to S^2$, every two fibers are linked with linking number one. [duplicate]

Regard $S^2$ as copmlex projective plane. $$S^3= \{(z_0,z_1)\in \mathbb{C}^2\mid z_0\overline{z_0}+z_1\overline{z_1}=1\}$$$$p: S^3\to\mathbb{C}P^1,\quad p(z_1,z_2)=[z_1,z_2].$$ Suppose that $z_1\neq ...
knot's user avatar
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Alternative Geometric Construction of Minkowski Space Metric

In the Minkowski Space, at each point, we define a tangent space, and using the metric construct, we calculate a real number that shows the infinitesimal invariant interval between two points. This ...
VVM's user avatar
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Why $K_\nabla$ is the same as $K_{\omega_S}$?

Let $G\subset GL_n(\mathbb{R})$ be a Lie subgroup and let $M$ be a manifold. Consider $S$ a $G-$structure on $M$ and $\nabla$ a connection compatible with $S$. We know that if $E=E(P,V)$ is the vector ...
Armando Patrizio's user avatar
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2 answers
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On a definition of covering space in terms of fiber bundles

I'm currently taking a course in (smooth) fiber bundles. I am not very used to covering spaces, but I'm trying to understand them in the language of fiber bundles. I would like to know whether the ...
Níckolas Alves's user avatar
5 votes
2 answers
347 views

Flat Bundle vs Trivial bundle

In R.W. Sharpe's Differential Geometry, a flat fibre bundle is defined as a bundle whose transition functions are constant. I don't understand the difference between this and a trivial bundle. Because,...
Jarah Fluxman's user avatar
2 votes
1 answer
64 views

The equivalence relation when constructing the associated bundle

When constructing from a typical fibre $F$, an action of a Lie group $G$ on that fibre and a $G$-principal bundle $P$ over some base space $M$ the associated bundle $P[F]$, one does this by ...
Jannik Pitt's user avatar
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1 answer
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When does the map from Fundamental Group to Holonomy Group Injective?

We know that over a rank $n$ vector bundle $E$ (with base space being $M$, a manifold), if the connection $\nabla$ is flat, then the parallel transport along loop $\gamma:I\to M$ for $E$ will only ...
BoyanLiu's user avatar
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Definition of the Principal Bundle for smooth manifolds?

The fiber bundle is defined as A fiber bundle is defined as the tuple $(E, B, \pi)$ where $\pi: E \to B$ is a continuous surjective map from topological space $E$ to topological space $B$. ...
Jagerber48's user avatar
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4 votes
1 answer
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What kind of notation is $\pi: b\ \varnothing\ M$?

In the paper Magnetic Bloch analysis and Bochner Laplacians of Ruedi Seiler, this expression appears in this context: The conventional notation for a fiber bundle would be $\pi: b\ \to\ M$. where $\...
mma's user avatar
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1 answer
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Generalizations of fibre products

For maps $f:X\to Z$ and $g:Y\to Z$ of topological spaces, we can define fibre product as $X\times_{Z}Y=\{(x,y)\in X\times Y \mid f(x)=g(y)\}$. I was wondering if there is a generalization of this ...
Alexander93's user avatar
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Is this is valid/standard definition of a principal bundle?

I'm having a hard time understanding principal bundles. It seems there are a lot of definitions around which is making it even more confusing for me. Is this a valid definition of a principal bundle: ...
Jagerber48's user avatar
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1 answer
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Difference between $G$-principal bundle and fiber bundle with fiber $G$?

I'm having a hard time understanding principal bundles. I understand that a fiber bundle can be defined $(E, B, \pi, F)$ where $E, B, F$ are topological spaces and $\pi: E\to B$ is a surjection. For ...
Jagerber48's user avatar
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1 vote
1 answer
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Is there always a bijection between a fiber bundle and the cartesian product?

It's well known that there are non-trivial bundles such as the Mobius strip. In these cases it is known the bundle $E$ with bundle map $\pi : E \to B$ over base space $B$ with fiber $F$ cannot be ...
Jagerber48's user avatar
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0 answers
48 views

Existence of principal G-bundle given an associated vector bundle

I am wondering if the following is true. Let $G$ be a Lie group, $V$ a vector space, $\rho$ a representation of $G$ on V, and $\pi: E\rightarrow M$ a vector bundle with fibre $V$. Does there exist a ...
Flo's user avatar
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4 votes
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When is a PDE a subbundle of a jet bundle (as opposed to a fibered submanifold or just a closed embedded submanifold)?

Nowadays it is common to see PDEs defined as either closed embedded submanifolds of a jet bundle (of some appropriate order) or else, if some further conditions are satisfied, as a fibered submanifold....
Jim's user avatar
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1 vote
0 answers
37 views

Existence of holomorphic vertical vector fields

Let $X$ and $Y$ be complex manifolds and let $\pi \colon X \to Y$ be holomorphic proper surjective submersion. Therefore, $\pi$ is a fiber bundle whose fibers are complex manifolds. My question: Does ...
KuSi's user avatar
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1 vote
1 answer
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Error in Hatcher, Algebraic Topology?

In Hatcher's Algebraic Topology on page $409$ in the third paragraph it is written Given a fibration $p : E \to B$ with fiber $F = p^{−1} (b_0 )$, we know that the inclusion of $F$ into the homotopy ...
psl2Z's user avatar
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0 votes
1 answer
35 views

Fibre bundle over Borel subgroup with fibre the complete flag

Let $G = \mathrm{GL}_n$, let $B$ be a Borel subgroup of $G$. The set $\mathcal{B}$ of $G$-conjugates of $B$ can be given an algebraic variety structure and is known as the variety of Borel subgroups ...
Gutiérrez's user avatar
2 votes
0 answers
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Computation of homology groups of Milnor's exotic sphere

Milnor's sphere $M$ is defined as the total space of the $S^3$ fiber bundle over $S^4$ with clutching map $f : S^3 \to SO(4)$ given by $u \mapsto (x \mapsto u^ixu^j)$, where $i, j$ are constants and $...
pnguyen0719 's user avatar
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1 answer
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$Z$ definition? (Hussemoller, fibre bundles)

Hussemoller doesn't describe what $Z$ is in this definition (page 67, third edition): Note that $\mathbf L(\mathbb F^m, \mathbb F^n)$ is the collection of all linear functions $\mathbb F^m \...
Nate's user avatar
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0 answers
44 views

How to show that the map $E \times I \to E: (v,t) \to t\cdot v$ is continuous on a vector bundle $E$?

Let $E$ be a vector bundle over a topological space $M$. I want to show that $\pi:E \to M$ is a homotopy equivalence. To show this, I use the zero section $\zeta:M \to E$. Then $\pi \circ \zeta = Id_M$...
nomadicmathematician's user avatar
1 vote
2 answers
122 views

Determinant of endomorphism bundle

We have that the endomorphism bundle of a smooth vector bundle $E$ is $\text{End}(E) = \text{Hom(E,E)}$. Is it necessarily true that $\det(\text{End}(E))$ is always trivial for any smooth vector ...
Jeff's user avatar
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0 answers
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Dual bundle transition functions

We can show that the dual bundle $E^*$ with fibers $(E^*)_x = (E_x)^*$ for all $x \in B$ have transition functions $g_{\alpha\beta} = (f_{\alpha\beta}^T)^{-1} : U_\alpha \cap U_\beta \to GL(r,\mathbb ...
Jeff's user avatar
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0 votes
1 answer
61 views

Right G-Space (Husemoller, Fiber Bundles)

Hussemoller's definition of a right G-space confuses me at a few key points: For a topological group G, a right G-space is a space $X$ together with a map $X \times G \xrightarrow{\quad }X$. The image ...
Nate's user avatar
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1 vote
1 answer
43 views

B-Isomorphic? (Husemoller, Fiber Bundles). Illustrated diagrams included.

Wikipedia gives the definition of a bundle map as the arrow $\left( \xrightarrow{\quad \varphi \quad } \right)$ in the commuting diagram below: Hussemoller gives the definition of a bundle map as the ...
Nate's user avatar
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0 answers
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If Chern classes are only defined for vector bundles, why can a $U(1)$ principal bundle have associated Chern classes?

When people talk about Chern classes of $U(1)$ principal bundles, do they really mean the Chern class of the associated vector bundle? As far as I can understand, Chern classes exist only for complex ...
aQuarkyName's user avatar
2 votes
0 answers
62 views

Discrete cocycle datum of a principal $G$-bundle

Let $X$ be the topological realization of a finite simplicial complex, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. Let's recall the standard fact that more generally for any numerable ...
JackYo's user avatar
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3 votes
0 answers
88 views

Gysin sequence from Serre spectral sequence vs Thom isomrophism

Let $n\geq 1$ and let $\pi:E\to B$ be a fiber bundle with fiber $S^n$ with $B$ simply connected. We will fix an orientation of $p$. An easy analysis of the Serre spectral sequence shows that there is ...
Ken's user avatar
  • 2,622
1 vote
2 answers
100 views

"Open" Cell? (Hatcher and Husemoller)

Have $\mathbb D^n$ be some n-dimensional unit disk. Have $\partial S$ denote the boundary of some space $S$. Let infix $(-)$ denote a difference between collections of objects. In Dale Husemoller's ...
Nate's user avatar
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0 answers
32 views

Are function spaces over a shrinking set vector bundles?

I have function spaces $F(S_t) \subseteq \{f : S_t \to \mathbb{R}\}$ defined over shrinking sets $S_t\subset S_s$ for $t> s$. I have a trivial fiber bundle $[0,\infty) \times F(S_0)$ where the ...
lightxbulb's user avatar
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1 vote
0 answers
71 views

$\pi$ is a fiber bundle over $B$... Difficulty interpreting some Wikipedia definitions

Let (i)$\left(F \xrightarrow{\quad \quad} E \xrightarrow{\quad \pi \quad}B \right)$ be any fiber bundle. Wikipedia has a different parlance when defining some fiber bundles. Take this excerpt from the ...
Nate's user avatar
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1 vote
1 answer
155 views

How can de Rham cohomology find obstructions?

A differential form is closed iff its exterior derivative is $0$. A differential $k$-form $w$ is exact iff there exists a differential $(k-1)$-form $\eta$ so that $\hbox{d}\eta = \omega$. Every exact ...
étale-cohomology's user avatar
3 votes
1 answer
50 views

Requirements for trivial fiber bundles

I am starting to learn about fiber bundles, and wanted to understand the following. We consider a fiber bundle $(E,B,\pi,F)$ where $E,B,F$ are smooth manifolds, and $\pi : E \to B$ is a continuous ...
Jeff's user avatar
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3 votes
0 answers
102 views

Flat Maurer-Cartan connection iff flat Berry connection

I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$. The first is the canonical or $H$-...
Victor V Albert's user avatar
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35 views

How to find the cocycle and sections of the covering $\pi:S^{1}\rightarrow S^{1}$ such that $\pi(z)=z^{2}$ with $F=\mathbb{Z}_{2}$

In differential geometry we reached the topic of cocycles and sections, but we only covered the definitions. Due to this I am trying to solve some problems of a given study problem list, so I can ...
Roberto Dias culebro's user avatar
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1 answer
107 views

Equating two definitions of principal fiber bundles

I am following these lectures on principal fiber bundles. Here, a principal fiber bundle is defined as a fiber bundle of which total space $P$ has a right free action of some Lie group and which is ...
Lourenco Entrudo's user avatar

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