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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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What is Thom Isomorphism?

I am reading the following post on Thom Isomoprhism and I also have the Thom Isomoprhism from Hatcher's, Corollary 4.9,pg441 nlab's: Let $V \rightarrow X$ be a rank $n$ vector bundle over a simply ...
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Is the map $\bigsqcup_{w } \frac{BwP}{P} = \frac{GL_n(\mathbb{C})}{P} \to GL_n(\mathbb{C})$ via $bwP \mapsto bw$ a morphism of algebraic varieties?

Let $G = GL_n(\mathbb{C})$, $P \subseteq G$ a parabolic subgroup, and consider the decomposition $\frac{G}{P} = \bigsqcup_{w \in W^P} \frac{BwP}{P}$ where $W \cong S_n$ is the Weyl group of $G$, $W_P$ ...
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Classifying maps and transition functions

Suppose a space $X'$ is obtained from $X$ by attaching an $i$-cell, i.e., $$X' = X \cup_\phi e^i$$ where $\phi: \partial e^i \to X$ is the attaching map. Let $G$ be a structure group, and $P$ be a ...
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Change of trivialization

Let $\pi:P\to M$ a smooth $G$-principal bundle with action $\beta:G\times P\to P$, $G$ a Lie group and $M$ a diff. manifold. Let also $T^*P$ be its cotangent bundle and $\sigma$ be a section of this ...
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Principal fibre bundles with constant transition functions

I guess this is not limited to principal bundles, however those are my primary interest in asking this question. Let $(P,\pi,M,G)$ be a principal fibre bundle over $M$ with structure group $G$. ...
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1answer
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Classification of circle bundles over $\mathbb{RP}^2$

I am trying to understand isomorphism classes of bundles $$\mathbb{S}^1\hookrightarrow E\to \mathbb{RP}^2.$$ These are classified by homotopy classes $[\mathbb{RP}^2,BAut(\mathbb{S}^1)]$. ATTEMPT 1. ...
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If $G$ is a Lie group and $H$ is a closed Lie subgroup, then $G\to G/H$ is a principal- $H$ bundle.

Let $G$ be a Lie group and $H$ be a closed Lie subgroup of $G$. Let $G/H$ has the quotient topology. Then $ p: G\to G/H$ is a principal-$H$ bundle. I was reading the above theorem from the book ...
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Fibre bundles over interval are trivial - but in a smooth way?

I'm trying to understand the homotopy invariance theorem for smooth fibre bundles. A main step is to show that the interval $[0,1]$ only admits trivial fibre bundles. I found this proof (see Lemma 3) ...
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induced map in homology on a fiber bundle

Let $F \rightarrow E \rightarrow B$ be a fiber bundle of compact connected smooth manifolds and $B$ simply connected. Suppose that there is a map $f: E \rightarrow E$ that covers a map $g: B \...
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Reference Request for $\frac{G}{B} \rightarrow \frac{G}{P}$ being a fiber bundle for $G = GL_n(\mathbb{C})$

Let $G = GL_n(\mathbb{C})$, let $B$ be the Borel subgroup of upper triangular matrices, and let $P$ be a parabolic subgroup. Then we may identify $G/B$ with the complete flag variety and $G/P$ with ...
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Proof fiber bundle construction theorem

Do you think this proof is correct? https://en.wikipedia.org/wiki/Fiber_bundle_construction_theorem Have you got any further reference?
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Jet bundle cohomology

Consider a base manifold $\mathcal{M}$ and a smooth bundle $E\to\mathcal{M}$. I am interested in the cohomology groups of the variational bicomplex associated with the jet bundle $J^{\infty}E$. In ...
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Construction of fiber bundles via transition maps: how to find smooth structure.

I'm trying to prove the reconstruction theorem for general smooth fiber bundles, that is, no extra algebraic structure over my bundle, only smoothness. Let $M$ and $F$ be smooth manifolds of ...
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2answers
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Monodromy when fiber is connected

When I learned about monodromy, it was in the context of covering spaces. There, if $p$ is the covering map and $l$ is a loop with $\gamma(0)=\gamma(1)=x$, then a $\bar \gamma$ lift of $\gamma$ is ...
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parallel transport is independent of the bases chosen in each of the two tangent spaces

Connections on principal fibre bundles In the above set of notes on page-3 section 2.2 under the heading Parallel transport there is a statement that equivariance ensures that the parallel transport ...
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1answer
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Consequences from Bott Periodicity

I have some questions about some arguments used in the discussion about consequnces of Bott periodity in in A Concise Course in Algebraic Topology by P. May at page 207. Her the excerpt: Following ...
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Are most physics books wrong about the covariant derivative and connection?

I have always read in many physics books that a valid way of intuitively introducing the covariant derivative and the connection was the following: (example in GR but same thing for gauge theories) ...
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2answers
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G-structure on fiber bundles

I was studying about fiber bundles with a $G$-structure, and I arrive to the definition (below all the spaces are smooth manifolds): Given a topological group $G$ a $G$-structure on fiber bundle $(E,...
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Problem with equivalent definition of a integrable $G$-structure

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 2. It this proposition: My problem is that I don't understand the converse of this ...
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1answer
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Section and pull-back bundle

I find in general that the pull-back of a section of a vector bundle is a section of the pull-back bundle, but this seems to be false for the cotangent bundle. Let $\phi:M\to N$ be a smooth map, $\...
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Pulling back $\mathfrak g $-valued 1-forms

Let $\omega$ be a vector-valued 1-form where the vector space is the Lie algebra $\mathfrak g$ and let $\mathcal P$ be a trivial principal bundle over a manifold $M$. Thus $$ \omega\in\Omega^1(\...
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1answer
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Tangent bundle of a trivial bundle

I was asking myself if the tangent bundle of a trivial bundle $\mathcal{P}=M\times V$ with fiber $\pi: \mathcal{P}\to M$ (actually it would be a principal trivial bundle, but I think it doesn't matter ...
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1answer
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Torus bundle over a torus

Let $G$ be a nilpotent Lie group and $H<J$ be two closed subgroups such that $$G/H\to G/J$$ is a fiber bundle with fibers $J/H$. Suppose that $G/J=(S^1)^m$ and $J/H=(S^1)^m$. Is the bundle trivial?...
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Contractible fibers implies $H^*(E) \cong H^*(B)$ for $p:E \rightarrow B$?

I believe the following is true: Let $F\rightarrow E \rightarrow B$ be a fiber bundle. If $F$ is contractible, then $H^*(E;R) \cong H^*(B;R)$ for a commutative ring $R$ with identity. I will ...
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1answer
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Question about group of automorphism of some $G$-structure.

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 15. I don't understand why $U$ consists of transformation $a$ of $M$ that leave each ...
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1answer
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Can we detect non-trivial bundles using characteristic classes for a custom structure group?

Suppose $E\to X$ is a fibre bundle where say $X$ is a CW complex, with structure group $G$, i.e. it is classified up to bundle isomorphism by a homotopy class of maps $c\colon X\to BG$. It's not ...
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1answer
28 views

Help find mistake in conclusion about vector fields on principal bundle

Let $P$ be a principal fiber bundle with structure group $G$ acting freely on the right. Let $T_uP = G_u + Q_u$ be a connection on $P$, where $u\in P$ and $G_u$ is the vertical space consisting of ...
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1answer
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Do the isomorphism classes of fiber bundles constitute a set?

Let $ M $ and $ F $ be smooth manifolds. Is the collection of isomorphism classes of fiber bundles of fiber type $ F $ over $ M $ a set or not and why?
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1answer
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Fiber Bundles over spheres

Let $F$ be any topological space. In many books, for example in http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html, it is said that a continuous map (characteristic map) (where $Homeo(F)$ has the ...
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1answer
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Is the Hairy Ball Theorem equivalent to saying that the Hopf Fibration has no global sections?

The Hairy Ball Theorem states that $S^2$ has no nonvanishing tangent vector fields. But if we did have such a field then we could normalise each vector so that it lay on the unit circle of the tangent ...
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1answer
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Local triviality for the fiber bundles

The notations are as follows: \begin{align*} & \operatorname{Diff}^+(\mathbb{D}^2):= \{ f:\mathbb{D}^2\to \mathbb{D}^2\ |\ f \text{ is an orientation preserving diffeomorphism} \},\\ & \...
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1answer
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The orbit space GL(n,R)/O(n)

If $G= GL(n,\mathbb{R})$ and $H= O(n)$ then why the orbit space $G/H$ is homeomorphic to the space of all upper triangular matrices with positive diagonal entries?(Here action of $H$ on $G$ is the ...
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Natural bijection between equivariant maps and sections, Principal bundles

This is in page 11, Prop 6.1 of Mitchell's notes on Fibre bundles. I am quoting the proposition below. Let $\pi:P \rightarrow B$ be a principal $G$ bundle, $X$ a right $G$-space. $Hom_G(P,X)$ ...
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2answers
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Doubt in the definition of principal bundle.

I am following Kobayashi and Nomizu( Foundations of differential geometry) Volume 1. In page number 50 while defining principle $G$-bundle $P(M,G)$ they said that the action of the Lie group $G$ ...
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1answer
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Question about a proposition in Kobayashi book about $G$-structures.

I'm reading Kobayashi's book, Transformation Groups in Differential Geometry and at the page 3 is this proposition: the definition of $K$ is given here: My question is why this proposition is ...
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1answer
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Extending a monomorphism of bundles, Lemma 7.3, Atiyah, Shapiro

This is from Lemma 7.2, pg17 Let $E,F$ be vector bundles on $X$ and $f:E \rightarrow F$ a monomorphism on $Y$. Then if $\dim F > \dim E+ \dim X$, $f$ can be extended to a monomorphismon on $X$ ...
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Definition of fibration from a compact Kahler manifold to a compact complex curve

Let $X$ be a compact Kähler manifold and let f be a holomorphic map from $X$ to a compact complex curve $C$. We say that f is a fibration if it is surjective and if its generic fiber is connected. I ...
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Must every manifold bundle be a fiber bundle?

I have been learning differential geometry from this lecture series by Frederic Schuller's. Here he defines a bundle (of topological manifolds) as A triple, $(E, \pi, M),$ where $E$ and $M$ are ...
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$H$ is admissible, then $P\rightarrow P/H$ is a principle $H$ -bundle.

Proposition 3.5, page 5: Suppose $P \rightarrow B$ is a principal $G$-bundle, and let $H$ be an admissible subgroup of $G$. Then the quotient map $P \rightarrow P/H$ is a principal $H$-bundle. ...
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1answer
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Surjective homotopy equivalence which is not a fibration?

This is probably obvious to topologists so I'll just come right out with the question: What is an example of a surjective homotopy equivalence $E \to B$ of path-connected CW complexes which is not a ...
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1answer
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Are there topologically trivial bundles with a nonzero curvature?

A famous example of a topologically nontrivial bundle is the Moebius strip which is a nontrivial bundle over the circle. A topological trivial analogue would be a cylinder. Is it possible to have a ...
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0answers
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A theorem on fibre bundles, $H^k(E) \cong H^{k+n}(E,E_0)$.

This is a theorem about fibre bundles, stated in page 21, Theorem 3.6 Let $\xi= (E,B,p)$ be an $n$-vector bundle. Let $F_0$ be the fibre $F$ without its nonzero element, and $E_0$ be the total ...
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differential of integration over fibers

Is there a way to express the differential of a fiber integral that is similar to Reynolds transport problem or Leibniz rule? Here is the following setting: Let $\pi: X\mapsto Y$ be a fiber bundle on ...
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1answer
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Integral curves are horizontal

Suppose we're given an Ehresmann connection on a submersion (or a fiber bundle, but I don't think it's needed) $\begin{smallmatrix}X\\ \downarrow\\ Y \end{smallmatrix}$. Given a curve $\gamma$ in the ...
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Descent data and trivialization of bundles via coherent isomorphisms of fibers

In this MO question I tried to understand how a trivialization of a bundle (continuous map) $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$ is related to a coherent family of isomorphisms ...
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Infinitely many fixed-rank bundles on projective space?

I was asked as a question whether a topological space can have infinitely many non-isomorphic bundles (with fixed rank). As a hint we were recommended to consider projective space. I don't really know ...
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1answer
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Nontriviality of the Hopf Fibration

A simple question how to understand why even though locally $S^3$ is homeomorphic to $S^2\times S^1$, how do you see that globally this is not true?
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1answer
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Is there a group whose manifold is a fiber bundle with base is $S_1$ and fiber $\mathbb{Z_2}?$

Let's consider a fiber bundle with base $S_1$ and fiber $\mathbb{Z}_2$. I want this manifold to be topologically non-trivial, the edge of the Möbius strip. How do I know if is it possible to ...
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Universal standard principle bundle of the Gauge group

Let $p:P\rightarrow B$ be a principal $G$-bundle, and let $E_G\rightarrow B_G$ be the universal bundle for $G$. Let $Aut_B(P)$ be the subspace of $Map_G(P,P)$ of maps over $B$. Let $Map_P(B,B_G)$ be ...
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Is a union of compatible fiber bundles a fiber bundle?

Let $\eta_i:E_i\rightarrow B_i$ be fiber bundles, and let $f_i:\eta_i\rightarrow\eta_{i+1}$ be maps of fiber bundles. Assuming that the bundles maps $\eta_i$ are maps of smooth compact manifolds, and ...