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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.

In mathematics, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.

Specifically, the similarity between a space $E$ and a product space $B × F$ is defined using a continuous surjective map: $\pi \colon E \to B$ that in small regions of $E$ behaves just like a projection from corresponding regions of $B × F$ to $B$. The map $π$, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space $E$ is known as the total space of the fiber bundle, $B$ as the base space, and $F$ the fiber.

1233 questions
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Compactness about fiber bundle

I am working on a problem (problem 10.19 (d)) in John M. Lee's Introduction to Smooth Manifold. Assume that $\pi$: $E$ $\rightarrow$ $M$ is a fiber bundle with model fiber $F$, I need to prove if $E$ is compact, then so are $M$ and $F$. Clearly, $M$…
Yunmath
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Whitney sum of Mobius bundles

I'm currently working through Nakahara's book and I've hit a snag on exercise 9.1. Consider the real line bundle $ L $ of $ S^1 $ $ \left ( L,S^1,\pi,\mathbb{R}, G \right ) $ $ L $ is either the cylinder bundle or the Möbius bundle. Consider the…
Seenathin
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Example of non global section

I am very new to differential geometry. I am familiar with fibered manifolds, fibered bundles (i.e fibered manifold with local trivialization) and sections. No I want to motivate the existence of local sections and for that I am looking for an…
JDoe
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Do all fibre bundles have a structure group?

The transition functions of a vector bundle over the field $\mathbb{F}$ are in $GL(n,\mathbb{F})$. Such a vector bundle has structure group equal to a a subgroup of $GL(n,\mathbb{F})$. Do all fibre bundles have a structure group? If the transition…
Timon van der Berg
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Question about fiber bundles

The following is from page 14 of the book heat kernels and Dirac operators Definition: let $\pi: \mathcal{E} \rightarrow M $ be a smooth map from a manifold $\mathcal{E}$ to a manifold $M$. We say that $(\mathcal{E},\pi)$ is a fiber bundle with…
Asma
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Classifying the twisted I-bundle over Klein bottle

I wanted to know how can I prove there are only two twisted I bundle over the Klein bottle. As I looked some topology texts, I couldn't find any solid definition of twisted I-bundle (or general definition of twisted fiber bundle) except that it is…
siavash
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On the existence of some kind of "universal fibre bundle"

While attending an introductory course on the theory of (smooth) fibre bundles, an example I was given of (principal) bundles was that of topological coverings of a space with structure group the galois group of the covering. In this case, when the…
Ukitinu
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Is a fiber bundle uniquely determined by its holonomy?

Consider two fiber bundles with the same base space (with the same base point). If the holonomy of the two bundles for each loop from the base point to the base point are the same, does this imply the two fiber bundles are equivalent?
Xiao-Gang Wen
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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 $(E,M,F)$ with projection $\pi:E\rightarrow M$ and…
user418048
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Clarification on Fiber bundles

I'm a little confused about fiber bundles. I have a specific example and would appreciate someone clarifying this for me. Let $f:M\times U\rightarrow TM$ be a map from a product space $M\times U\rightarrow$ to the tangent bundle TM of M. Is…
Jorge
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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 is said that if $ E_1 $ is a fiber bundle over $E$ with a…
Amd
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Why is the stalk cohomology of the nearby cycles of a sheaf complex exactly the hypercohomology of the Milnor fiber of that complex?

Let $X$ be a complex manifold and $f\colon X\to\mathbb{C}$ a non-constant holomorphic map. We define $X_t=f^{-1}(t)$. Let $\mathcal{F}^\bullet\in D^b(X)$ and we denote by $\psi_f$ the nearby cycle functor and by $F_x$ the Milnor fiber of $f$ at…
Marco
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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 compact-open topology) $$\phi \colon S^{n-1} \to…
ychemama
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Why there can be different parametrizations?

I've been getting started with fiber bundles and I'm really confused up to now. The definition I have is clumsy and incorporates three things: local trivializations, transition maps and structure groups. It is in short like that: A fiber bundle is…
Gold
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Fiber bundle definition: Why is this a homeomorphism?

I'm getting started with fiber bundles and I have a doubt on the definition. I'll state just the part of the definition I'm in doubt: a fiber bundle is $(E,B,\pi,F,G)$ where $E,B,F$ are topological spaces, $G$ a topological group of homeomorphisms…
Gold
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