# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...