Tagged Questions

In mathematics, and particularly topology, 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. (Def: http://en.m.wikipedia.org/wiki/Fiber_bundle)

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Principal bundle as homotopy fiber universally self-trivializes

In this MO answer, I was told the definition of principal bundle as a homotopy fiber of its classifying map precisely says that it's the universal bundle which trivializes itself. However, I'm having ...
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Smooth structure in reconstruction theorem

Let $M,F$ be smooth manifolds, $\{U_i:i\in I\}$ an open cover of $M$ and a cocycle $\{t_{ij}:U_i\cap U_j\to\mathrm{Diff}(F)\}$. In almost any book which discusses fibre bundles, one can find the ...
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How do I see if the induced homomorphism from the inclusion map $S^{1'}\to S^1\times S^3$ is injective

Let $S^1\times S^3$ be parametrised as $\{(\alpha,\beta, \gamma)\in \mathbb{C}^3||\alpha|^2+|\beta|^2=1, |\gamma|=1\}$ and let $S^{1'}=\{e^{i\theta}(1,0,1)|\theta\in [0,2\pi]\}$. I would like to see ...
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Calculating The Fundamental Group of the Hopf Fibration

I want to calculate the fundamental group of the Hopf fibration (or rather, the fundamental group of the total space of the fibre bundle that is the hopf fibration). I know that $S^3$ is simply ...
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Difference between product projections and split epis in $\mathbf{Top}$

I don't understand the excerpt below. The trivial bundles were defined as split epimorphisms, and since the ground category was additive with kernels (in fact abelian), they are the same as ...
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Are continuous product projections always split?

Are continuous product projections always split? What's an example of a product projection without a continuous right inverse?
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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 ...
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Fiber bundles with varying fibers via pullbacks along étale surjections

Suppose $\mathsf C$ is a complete extensive category. I managed to prove that a bundle $\alpha$ is a fiber bundle with fiber $F$ if there exists an associated cover $p:\coprod_iU_i\rightarrow B$ such ...
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What defines a fiber bundle?

I am slightly confused by Geometry, Topology and Physics by M. Nakahara. The following definition of fibre bundle $(E, \pi, M, F, G)$ is given: $E, M, F$ are differentiable manifolds, $E$ being the ...
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Every fiber bundle with Cantor set fiber is the suspension of a homeomorphism of the Cantor set.

I've heard that every fiber bundle (over $\mathbb S^1$?) with Cantor set fiber is the suspension of a homeomorphism of the Cantor set. Can someone explain the intuition behind the fact? Is there a ...
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Smoothing a continuous section in a fibre bundle.

Let $\pi: X \rightarrow M$ be a smooth fibre bundle and let $p^{1}_{0} : X^{(1)} \rightarrow X$ be its 1-jet bundle. Suppose there is a $\mathcal{C}^{1}$ section $h: M \rightarrow X$ such that it is ...
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Euler classes of oriented $2$-dimensional vector bundle, oriented $S^1$-bundle same?

As the question title suggests, are the Euler classes of an oriented $2$-dimensional vector bundle and of an oriented $S^1$-bundle the same?
What is $P\times_G E$?
I know what is principal bundle and associated bundle according Wiki.But I am not understand what is $P\times_G E$ .Seemly it is bundle,but I am not sure what structure is it . Below picture is from ...