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.
1,235
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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 ...
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Fiber-preserving diffeomorphism
What does it mean in the definition 7.1(ii) that the diffeomorphism
$$\phi_U$$ is fiber-preserving?
In what sense/how technically it preserves fibers?
They have never defined this in the book before.
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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 ...
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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 $B$. ...
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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 $\...
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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 ...
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Difference between a function and a section of a fibre bundle
Suppose $E \rightarrow B$ with projection map $\pi$ and fibre F is a fibre bundle, with a section $\sigma$. How is $\sigma$ different from a function $f:B \rightarrow F$? The standard answer I find ...
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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 ...
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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:
...
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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 ...
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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 ...
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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....
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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 ...
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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 ...
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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 ...
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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 $...
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$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 \...
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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 $\pi \circ \zeta = Id_M$...
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Pushforward, tangent map, vertical endomorphism and all that
I've already posted a similar question a couple of times, here and here, without receiving a fully clarifying answer, so I post it again, trying to be more specific.
Suppose you have a smooth ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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"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 ...
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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 ...
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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 ...
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$\pi$ 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 ...
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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 $H$-...
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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 ...
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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 ...
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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 ...
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Principal G-bundle induces a homeomorphism between orbit space and base
I am reading Chapter 14 of the algebraic topology book by Tammo tom Dieck. However, it is not going smooth since the very beginning. I will give the definitions first:
Def (Principal G-bundle). Let $G$...
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Does the monodromy action of a fiber bundle lie in the bundle's structure group?
Suppose we have a (topological) fiber bundle $p:E\to B$ with fiber $F$ and structure group $G$. Since $G$ acts on $F$ by homeomorphisms, it induces an action on the (integral) homology $H_*(F)$, i.e., ...
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Does the associated bundle functor have left or right adjoints?
Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and $\bar{\...
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Are principal fibrations the same as group bundles?
I am reading Hatcher's algebraic topology, where he defines a fibration $F\to E\to B$ to be principal if up to choices of homotopy equivalences it can be written as $\Omega B'\to F'\to E'\to B'$ (with ...
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Hatcher algebraic topology 4.3.10 [duplicate]
The exercise seeks to define for a fibration $F\to E\to B$ an action of $\pi_1(E)$ on $\pi_n(F)$, using the homotopy lifting property. However the basic idea imitating the action by deck ...
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Nowhere vanishing section on the Mobius strip
This is similar to another question asked on here, but the answers there didn't help me (and it is a slightly different set-up).
Consider the vector bundle $$E := ([0,1]\times \Bbb{R})/\sim \to S^1$$ ...
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Does principal $O(n)$-bundle have to consist of orthogonal frames?
It seems that it is OK that principal $O(n)$-bundle be constructed from non-orthogonal frames. For example, $(E,p,M)$ is a vector bundle and $P$ is the frame bundle. Then smoothly at each point $x$ of ...
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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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Construction of sections of an algebra bundle
A section of a locally trivial algebra bundle (E,p,B) is a continuous map $S$ from $p:B\to E$ such $p\circ S= ID$. For an ideal bundle $H$ of $E$, I am trying to construct sections of $E$ such that ...
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Every smooth fiber bundle admits an Ehresmann connection?
Is it true that every locally trivial fibration(fiber bundle) in smooth category admits an Ehresmann connection? The converse is obviously true, and it's one of the conclusion of the Ehresmann's ...
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Atlas in Differential Geometry: one of the transitions functions is wrong
In the book Connections in classical and quantum field theory by L. Mangiarotti, G. Sardanashvily
I have a problem with $(1.1.3)$ formula below:
$$(x,v)\mapsto (x,\rho_{\xi\zeta}(x,v));$$
namely that ...
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Fiber bundle over fiber bundle is a fiber bundle
I am trying to understand why a fiber bundle over a fiber bundle is a fiber bundle (called composite). This appears in the book "Differential geometric structures" of Poor, p. 9.
There, it ...
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Global structure of submanifolds transverse to fibres of a fibre bundle
I try generalising the following "classical" result to the abstract fibre bundle (Rudolph-Schmidt, Differential Geometry and Mathematical Physics part I)
I can state it as follows.
Global ...
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Fiber bundles maps and Product base
Assume $f:E\rightarrow X\times [a,b]$ is a fiber bundle. Assume $X\times [a,c]$ and $X\times [c,b]$ for some $a<c<b$ are both trivial bundles. Let $h_1:f^{-1}(X\times [a,c])\rightarrow X\times [...
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Notation in the local form of a covariant derivative
Let $\pi: P \rightarrow M$ be a principal $G$-bundle with connection $A$ and $E = P \times_\rho V$ an associated vector bundle to $P$ where $V$ is a vector space and $\rho: G \rightarrow GL(V)$ is a ...
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Fiber bundles, orientation preserving map and pushforwards
I am reading an article about the geometric foundation of Hamiltonian Markov Chain Monte Carlo (see https://projecteuclid.org/journals/bernoulli/volume-23/issue-4A/The-geometric-foundations-of-...
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