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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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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Examples of points in the total space of a fiber bundle

What are some examples of notation for points in the total space of a fiber bundle? I think I have found one variant for vector bundles in this question about trivializations (something like $v^i \...
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1answer
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Fiber from $S^1$ to $\mathbb{R} P^1$ that does not admit a section.

Consider the map $q:S^1 \to \mathbb{R} P^1$ given by $v \mapsto \mathbb{R} v$. I want to show that this map does not admit sections. I've previously shown that $S^1$ is diffeomorphic to $\mathbb{R} P^...
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Fiber bundle whose fibers are fiber bundles

Suppose $F \rightarrow E \rightarrow M$ is a fiber bundle. I've been considering situations where $F$ can be identified as a product space, but started thinking that it's not necessary for the ...
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30 views

Regarding singular points of a fiber bundle

Let $X$, $Y$ be two projective varieties over $\mathbb{C}$, where $Y$ is smooth, and let $f:X\rightarrow Y$ be an etale locally trivial fiber bundle, with fiber a variety $F$. Is there a relation ...
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1answer
37 views

Fiber bundle $S^{3} \to S^{7} \to \mathbb{H}P^{1}$

The map $h: S⁷ \to HP¹ $ given by $(a,b,c,d) \to \frac{a}{c}$ is a fiber bundle with fiber $S^3$ over basepoint $\infty=\frac{a}{0}$. Thus, $h$ induced a sequece $$S^3 \cdots S^7 \to HP¹$$ Even more, ...
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1answer
38 views

Why doesn't the Homotopy group satisfy excision?

I'm studying higher homotopy groups from the book Algebraic Topology by author Allen Hatcher, there he says that the sequence $A \to X \to X/A$ does not induce an exact sequence of homotopy groups. ...
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subbundles & slice charts

Let $F\hookrightarrow E \overset{\pi}{\longrightarrow} M$, $F'\hookrightarrow E' \overset{\pi'}{\longrightarrow} M'$ two smooth fiber bundles. If $F'\subset F$, $M'\subset M$, $E'\subset E$ are ...
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About theorem 16.2 in Switzer's Algebraic Topology

I have some difficulties understanding a very precise point of Switzer's proof of the existence and unicity of chern classes, which is Theorem 16.2 in his book. Unfortunatly there are many notations ...
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Lens space bundles over a circle must come from sphere bundle over a circle?

The question I would like to answer is the following. From the classification of sphere bundles we know that the only orientable $S^3$ bundle over $S^1$ is $S^3 \times S^1$. So suppose we have lens ...
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Principal bundle over another principal bundle again principal?

I want to expand upon an earlier (partially answered) question: Is a principal bundle of a principal bundle still principal? As in said question, let $M, P_1, P_2$ be manifolds, let $P_1 \overset{\...
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1answer
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Nontrivial bundle in physics

A student of mine asked me why we are studying bundles in theoretical physics, and why it isn't enough to look at product spaces. I gave the example of the tangent bundle of the sphere, and of the ...
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connection from covariant derivate

Given a vector bundle $E$ with covariant derivate $\nabla:\mathfrak{X}(M)\times \Gamma E \longrightarrow \Gamma E$, an operator that $\forall g,h \in C^{\infty} M \;\; U,V \in \mathfrak{X}(M) \;\; X,Y ...
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Interesting examples of $Q$-fiber bundles

Let $X$ be a topological space. What are some interesting examples of $\mathbb{Q}$-fiber bundles (fiber bundles with fiber some $\mathbb{Q}$-vector space, though not necesarily vector bundles) on $X$? ...
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2answers
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How many $G$ bundles are there over a surface?

Let $G$ be a finite group, $M$ a 2-dimensional manifold. There are $$ \#\{ (a_1,b_1,a_2,b_2,\cdots,a_g,b_g) \in G^{2g}\,|\, \Pi_i[a_i,b_i]=1\} $$ many $G$ bundles over $M$ up to isomorphism, where $...
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2answers
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Stiefel manifold as a fibre bundle over another Stiefel manifold.

I want to show $V_n(\mathbb R^k)$ is a fibre bundle over $V_m(\mathbb R^k)$ were $n>m$ Here $V_n(\mathbb R^k)$ is the Stiefel manifold, that is the set of all $n$ orthogonal frames in $\mathbb R^k$...
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1answer
25 views

Fibre bundle which is not an open map [duplicate]

I am looking for a counterexample to the following claim: Let $p: E \rightarrow B$ be a fiber bundle, then $p$ is an open map. This is true when $E \simeq F \times B$ where $F$ is the fiber and $...
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Non-trivial $C^{k_1}$ vector bundle that is trivial as $C^{k_2}$ fibre bundle

For given $\infty \ge k_1 \ge k_2 \ge 0$, are there non-trivial $C^{k_1}$ vector bundles over a "sufficiently nice" space that are trivial as a $C^{k_2}$ fibre bundle? How nice can the space be and ...
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Induced Projective map of a vector $X$

I've been going through connections on fibre bundles in Nakahara's Topology, Geometry and Physics from 2003 and I wondered if someone could answer this question for me (from page 395, exercise 10.1 a.)...
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1answer
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Yang-Mills Field Strength Compatibility Function

I've been going through Nakahara's Topology, Geometry and Physics from 2003 and I'm struggling to fully understand this derivation from page 410: Say there exists two different fields, $\mathcal{F}_i ...
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1answer
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Higher covariant derivatives and the exterior derivative

Let me start with the following tl;dr version of my question What is a higher-order derivative, in general? How does it relate to the exterior derivative and to differential forms? Suppose we ...
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1answer
31 views

Oriented sphere bundles - Why does pulling back an orientation along a path respect the restriction map?

I am reading Edwin Spanier's book "Algebraic Topology", theorem 5.8.19. I do not understand the very first sentence of the proof. I hope there is not too much notational confusion here, but the ...
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1answer
46 views

The relationship between the tubular neighbourhoods of two diffeomorphic manifolds

I'm a beginner of this complex area and want to use the differential geometry as a tool to solve some control problems. So my statement might be a little bit inaccurate...I will try my best. There ...
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G-connections and holonomy group

$\pi:E \longrightarrow M$ vector bundle with connection $\nabla$ and structure group $G\leq GL(V)$. I read three definition of G-connection: 1) The holonomy group at $p$ are the parallel transports $...
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1answer
50 views

Tangent and cotangent bundle as an associated bundle

In a book I read the isomorphisms below were mentioned without any explanation. Is there any intuitive way to see these identities hold? Let $M$ be a smooth $n$-manifold, $F(M)$ a frame bundle of $...
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Adapted Stochastic Processes and Fiber Bundles

Suppose that we have an adapted stochastic process. In other words fix $T>0$ and suppose that for all $t \in [0,T]$ $X(t)$ is a collection of random variables, $F(t)$ is a $\sigma$-algebra, $F(s) \...
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Proving linearity properties of the curvature in a vector bundle

Let $\mathcal H$ be a connection on a vector bundle $E\overset{\pi}{\longrightarrow} M$. For $X \in \mathfrak X(M)$, let $\bar X \in \Gamma \mathcal H$ his horizontal lift, let $k(v):=\text{pr}_2(Vv):...
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1answer
58 views

Principal bundle structure of $S^1\to S^1$ given by $z\to z^2$ map

Let $S^1$ be unit circle on $C^\star$(punctured complex plane). Consider $S^1\to S^1$ by $z\to z^2$ map. Identify $Z_2$ action as $e:z\to z$ and $g:z\to -z$ action on $S^1$ of domain. Let $U=S^1-\{1\}$...
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1answer
63 views

2 dimensional torus bundles over $\mathbb{S}^2$

With my limited knowledge of bundles, it seems that the isomorphism classes of principal torus bundles are in one-to-one correspondence with the homotopy classes of maps $[\mathbb{S}^2,\mathbb{C}P^\...
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The definition of principal G-bundle

I've encountered many different definitons of principal $G$ bundle from Morita's Geometry of differential forms , Hamilton's Mathematical gauge theory , Kobayashi and Nomizu's Foundations of ...
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Do locally trivial and smooth fibrations correspond to subbundles of the tangent bundle

Let $M$ be a manifold of dimension $n$. Let $p:E\rightarrow M$ be a locally trivial smooth fibration. Does this give us a way to construct a subbundle of the tangent bundle $TE$ of the manifold $E$, ...
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1answer
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Existence of section on fiber bundles with contractible fibers.

I would like a proof (or a sketch of a proof or a reference) for the following fact: If $\pi \colon P \to M$ is a fiber bundle of manifolds, and its fibers are contractible, then there exist a ...
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1answer
54 views

Intuition of normal bundle to a manifold

So i recently learned about fibre bundles and tangent bundles in particular. While the definition of tangent bundles seems quite intuitive, i really struggle to understand any definition of the ...
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Fiber bundle maps

I was reading many books about fiber bundles and they don't agree in a general definition of a fiber bundle map. The categories I was trying to understand were "vector bundles", "principal bundles" ...
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1answer
97 views

The monodromy representation $\pi_1(X)\to Mod(F)$

Given a fiber bundle $F\to X\to E$, with $F,E$ are Riemann surfaces. I know the monodromy gives a permutation of the fiber. But how to see we have the monodromy representatio: $\pi_1(E)\to Mod(F)$? ...
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understand Torsion using Flat Bundles

My question refers to a discussion from this older thread on Neron-Severi group of a Kähler manifold. In the comments below Ted Shifrin's answer there arised a discussion when the map $H^2(X,\mathbb{...
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0answers
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May a Trivial Principal Bundle Admit a Non-Trivial Reduction of the Structure Group?

Let $\pi\colon P \to M$ be a principal bundle with structure group $G$ with respect to the Lie group right action $\triangleleft\colon P \times G \to P$. To my knowledge, a reduction of the structure ...
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1answer
174 views

$G$-bundles over $S^2$

In light of the fact that the only homogeneous (under a finite--dimensional Lie group action) $S^2$-bundle over $S^2$ is the trivial one. I would like to know if this fact is more general. i.e., is ...
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a little question about conjugate linear map

I'm reading Characteristic Classes written by Milnor and Stasheff,and the problem 13-F on page 153 says: If $U\subset \mathbb C^n$ is an open set with coordinate functions $z_1,...z_n:U\to \mathbb ...
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Fiber of the Hitchin fibration

In the paper "More On Gauge Theory And Geometric Langlands" by Edward Witten ( https://arxiv.org/abs/1506.04293 ), he writes about the Hitchin fibration $\pi: \mathcal{M}_H \rightarrow \mathcal{V}, (...
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1answer
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Homology with local coefficients and flat bundles

I have a problem in understanding Prop.1.10 in https://pages.uoregon.edu/sadofsky/691/sseq-local-coefficients.pdf My interpretation is as follows: Let $X$ be a smooth n-manifold and $U\to X$ be ...
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1answer
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A small question about fiber bundle [duplicate]

I'm recently studying fiber bundles,and I find many examples of them are motivated from maps like $p:E\to B$ with homeomorphic fiber $F=p^{-1}(b)$.Therefore I'm wonder if the converse is true,that is ...
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1answer
82 views

Short Exact Sequences as Fiber Bundles

I am very much a visual thinker, and so to me, to feel that I have understood a concept, it is rather important for me to be able to "see a picture of it in my head". Now I recognize that for many ...
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1answer
34 views

SCHEME-THEORIC FIBER

I am reading INTERSECTION THEORY (Fulton), I am in page 15 and I do not understand the meaning of : SCHEME-THEORIC FIBER (I know what is a fiber, but what does scheme-theoric means?
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Grassmann manifold

Grassmann manifold $G_{k,n}$ is the set of k-dimensional subspaces of $\mathbb{R^n}$. Let’s consider the set of k-frames $V_{k,n}$. I want to show that $$ G \to V_{k,n} \to^{\pi} G_{k,n} $$ can be ...
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1answer
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Is a Bundle whose Fibers are Homeomorphic a Fiber Bundle?

To avoid problems with the notoriously inconsistent notation/formalisms of geometry, I will define bundles and fiber bundles (as I call them) below: We define a bundle $E \xrightarrow{\pi} B \ $ as a ...
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1answer
98 views

Topology of fiber bundles

Everything is over $\mathbb C$. Let $X\to \mathbb P^1$ be a fiber bundle with fiber some smooth projective variety $F$. If we have another such bundle $Y \to \mathbb P^1$, then, for which $p,q$ we ...
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1answer
58 views

Why the pullback bundle is a submanifold?

Is $f^*E$ (the pullback of a smooth bundle) an embedded submanifold of $N\times E$? I know that it's well defined and at least immersed: Let $\pi: E \longrightarrow M$ a smooth fiber bundle and $f:...
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$\mathbb{P}^2(\mathbb{R})$ can't be a smooth fiber bundle of a $1$-dimensional smooth manifold

This question was stated during a lesson where I asked my professor if every two dimensional smooth manifold arises as a tangent bundle of some one dimensional smooth manifold. My professor said that ...
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Fiber bundles of $G$-spaces

So if $G$ is a topological group and $H,\ J$ are closed subgroups such that $H\lhd J$, then the principal bundle $G/H\to G/J$ is trivial iff it has a global section. I have questions about the ...

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