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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.

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Extending tangent vectors to commuting vector fields on a neighborhood

$M$ is a manifold, $X$ and $Y$ are tangent vectors in $T_xM$. Is it always possible to extend $X$ and $Y$ to local vector fields $U$ and $V$ around $x$ so that $[U,V]=0$ hold on some neighborhood of $x$? Or at least a weaker version: so that…
jw_
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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 $M$ choose a frame in the fiber $P_x$. Then at…
jw_
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How is $\pi^{-1}\left(U_i\right)$ different from $U_i \times F$ in fiber bundles?

On defining a fibre bundle, it is argued that the projection map $\pi$ requires to be satisfied the condition that there is a homeomorphism $h$ such that the first coordinate coincide. Definition (Fibre bundle). A fibre bundle structure on a space…
Eden Zane
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Prove that two fiber bundles are equal

Prove that $\tau(\Bbb{C}P^n)\oplus1=\eta\otimes\ldots\otimes\eta$ (n+1 times), where $1$ denotes the trivial complex line bundle over $\Bbb{C}P^n$, $\tau(\Bbb{C}P^n)$ is the complex tangent bundle of $\Bbb{C}P^n$ and $\eta$ denotes the Hopf bundle…
Austin
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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?
Roxana
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What is an algebra bundle?

If I associate a copy of an algebra with every point on a manifold - such that one could specify a connection between algebras associated with neighboring points - have I specified an algebra bundle? If the definition of an algebra bundle is more…
rossng
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Redefine fiber bundle

Can we redefine a fiber bundle only with transition function? How you are write trivialization of this? If you have a good reference please recommend to me to read it.
Ramtin.VA
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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 that the maps $f_i$ are inclusions of regular…
user09127
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Projection maps on (fibre) bundles

If one study bundles (being locally trivial or not, that does not matter) in the category of topological spaces, the projection map simply needs to be surjective or ''onto''. On the other hand, when one consider smooth manifolds, Wiki says that In…
Dog_69
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Fiber Bundle (Trivial Fibration)

Let $S^{1}$ be the unit circle with basepoint $1 \in S^{1}$. Consider the map $f_{n} : S^{1} \rightarrow S^{1}$ given by $f_{n}(z)=z^{n}$. Then $f_{n} : S^{1} \rightarrow S^{1}$ is locally trivial fibration with fiber a set of n distinct points (the…
LanaDR
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Diffeomorphism between local trivialization

I try to proof the following statement: Let $(W_p, F_p, t_p)$ and $(W_p', F_p', t_p')$ be two local trivialization around the same point $p \in M$, then $F_p$ and $F_p'$ are diffeomorphic. I showed that $(W_p \cap W_p') \times F_p$ and $(W_p \cap…
JDoe
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Principal bundle, horizontal and vertical subspaces

Given a principal bundle $P(M,G)$, we can decompose $$T_pP=V_pP\oplus H_pP.$$ I don't understand why $$[X,Y]\in H_pP$$ if $X\in H_pP$, and $Y\in V_pP$. Thanks!
Geo Rowe
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Transition maps of a principal bundle are smooth

A smooth principal fiber bundle is a smooth fiber bundle $\pi: E \to M$ together with a Lie group $G$ and a fiber preserving right action $E \times G \to E$ which restricts to each fiber freely and transitively. . How prove that the map $g_{\alpha…
user
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Notation clarification in fibre bundles

I am trying to comprehend the following material, I have some doubts mostly regarding the notations. Chern Classes: Throughout this section, we will use $\mathbb{Z}$ coefficients and let $\xi=(E, X, \pi)$ denote a $d$ 'dimensional vector bundle…
Eden Zane
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Is the following map a fiber bundle

Let $n$ be a positive integer. Let $O(n)$ be the orthogonal group. Let $I(k,n)$, for $0\leq k < n$, be the space of linear isometries $\mathbb{R}^k\rightarrow\mathbb{R}^n$. Is the map $O(n)\rightarrow I(k,n)$ given by restriciton a fiber bundle?…
user09127
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