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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Is a bundle morphism which restricts to homeomorphisms of the fibers a bundle isomorphism?

If $f$ is such a map between total spaces (assume a common base space) then it is a bijective and continuous and the inverse will be fiber preserving so that all we would need to prove is that the ...
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Cylinder as trivial fiber bundle with fiber $S^1$?

In prepartion for the example of a Mobius strip, a cylinder is often taken as a first example of a trivial fiber bundle with fiber $I$ and total space $S^1\times I$. However, it seems to me that we ...
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Are slices $\left\{b\right\}\times F\subset B\times F$ homeomorphic to $F$? [on hold]

Looking at a continuous projection $B\times F\rightarrow B$, are slices $\left\{b\right\}\times F\subset B\times F$ homeomorphic to $F$?
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Continuously variable *space* [migrated]

I'm trying to understand formally how and why a fiber bundle with fiber $F$ should be thought of as a gluing of homeomorphic copies of $F$ which varies continuously. I do not understand how this is ...
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Formalizing continuously indexed spaces in fiber bundles?

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle. As mentioned there, the point of local triviality seems to somehow ensure a ...
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Principal bundle isomorphism.

Let $G\longrightarrow P\overset{\pi}{\longrightarrow} M$ be a differentiable principal bundle, i.e. $M$ and $P$ are differentiable manifolds, $G$ is a Lie group, $\pi$ is a differentiable surjective ...
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Showing that the Hopf fibration is a non-trivial fibre bundle

I want to show that the Hopf bundle $$ \mathbb{S}^1 \rightarrow \mathbb{S^3} \rightarrow \mathbb{S}^2$$ is non-trivial as a principal fibre bundle. I have seen hints of several different approaches: ...
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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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What notation would I use to differentiate between a cartesian product and a cotangent bundle of surfaces?

If the $S^1$ is defined by $x^2 + y^2 = r^2$ , $T^2 = S^1 \times S^1$ is defined by $\left(\sqrt{x^2 + y^2} -R\right)^2 + z^2 = r^2$ , $T^3=S^1\times S^1\times S^1$ is defined by ...
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Line bundle trivial on fibers then isomorphic to the pullback of a line bundle

$\require{AMScd}$ I'm currently reading Milne's notes about Abelian varieties. On page 26 he proves the following theorem: Let $V$ and $T$ be varieties over $k$ with $V$ complete, and let ...
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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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How to lift an action along a covering

Took from Lawson-Michelsohn "Spin Geometry", pages 80-81. Let $E\to X$ be an oriented $n$-dimensional vector bundle over a manifold $X$, and let $P_{SO}E$ be the associated $SO(n)$-bundle. Assume ...
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Is there a Mobius (infinite) cylinder?

In order to understand the question of the title I need to understand another thing first. If we consider the Mobius band, locally, for a $U_i \subset S^1$, where $S^1$ is the base space, the bundle ...
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Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
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Is it possible for $R \oplus M$ and $R \oplus N$ to be isomorphic to each other if $M$ and $N$ are not isomorphic?

Suppose $M$ and $N$ are non-isomorphic $R$-modules (where $R$ is a commutative ring with a unit element).Can we conclude that $R \oplus M \not\simeq R \oplus N$ ? If not in this most general ...
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the definition of relative Thom spaces

I find the definition of Thom space on the book Characteristic classes, J.W. Milnor, J.D. Stasheff, page 205: For a vector bundle $\xi$ with a Riemannian metric over a $CW$-complex $B$, let $A$ ...
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Which constructions on vector bundles satisfy a universal property?

I'm wondering whether constructions on vector bundles, such as the Whitney sum or tensor product of two bundles, satisfy some kind of universal property. What I mean by "some kind of universal ...
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Why do we need the $S \otimes L^{1/2}$ bundle product to determine a $Spin_c$ structure?

I am reading Marino's book on topological field theory and 4-manifolds and I am very confused in the construction of the $Spin_c$ structures for manifolds that do not admit a $Spin$ structure. In the ...
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Construct a lifting from the the total space of a fiber bundle to its spoace of 1-jets

I am reading the book "Convex Integrtion Theory by D.Spring" and need some help. Let $\pi:X \rightarrow V$ be a smooth fiber bundle over a smooth base manifold $V$. Let $h\in\Gamma^{0}(X)$ and ...
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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?
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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 ...
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Coset spaces of a lie group and fiber bundles

I am reading Steenrod. He writes: Another example of a bundle is a Lie group $B$ operating as a transitive group of transformations on manifold $X$. The projection is defined by selecting a point ...
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Covering spaces as fiber bundles

I am reading Steenrod. I think there is something wrong or sloppy in his definition of a covering space as a fiber bundle. He writes: A fibre bundle consists of: (i) A topological space $B$ (ii) a ...
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Equivariant flat U(1) bundle on a torus

I am trying to understand equivariant flat $U(1)$ bundles on a torus, $(S^1)^N$. By equivariant, I mean equivariance with respect to the natural action of $(U(1))^N$ on $(S^1)^N$. $G$-equivariant ...
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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 ...
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Cocycle condition on bundles as some equivalence relation?

The three cocycle conditions on the transition maps of a fiber bundle look alot like the three condition defining equivalence relation - refelxivity, symmetry, and transitivity. I was wondering ...
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Vertical bundle of the pullback bundle

Let $\pi: E \rightarrow M$ be a smooth fibre bundle, $f \in C^{\infty}(N,M)$ for some smooth manifold $N$ and $f^{\ast}\pi: f^{\ast}E \rightarrow N$ the pullback bundle. How can I show the bundle ...
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Fiber bundle and fibration of classifying space

Let $BG$ is classifying space of $G$ topological group. If $G$ is any compact group and $H$ is a closed subgroup of $G$, then the inclusion map $i:H\rightarrow G$ induces \begin{equation*} ...
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Fibre bundle of Borel contruction for subgroups of compact groups

if $G$ is any compact group and $H$ is closed subgroup of $G$, then $G/H\rightarrow X_{H}\rightarrow X_{G}$ is a fibre bundle? ($X_G=X\times _{G}E=\left( X\times E\right) /G $ is orbit space where ...
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How to lift maps going into the base of a fiber bundle?

Let $p:E\to B$ be a fiber bundle and $f:B'\to B$ a map. Under what conditions does a lift $f':B'\to E$ exist? In the context of covering spaces, I remember a necessary and sufficient condition is that ...
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The square root of this operator is an isometry?

This question is motivated by the following proposition from the book "Connections, curvature and cohomology" by Werner Greub: If $(\xi, g)$, $(\eta, h)$ are riemannian vector bundles over the ...
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Covering maps as bundles

One geometric way to see a continuous map (or any set function really) is as a "fiber bundle" with the usual picture of a comb - the base space indexes the fibers of the map and there's a nice picture ...
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Action of fundamental group of circle on the homology of the fiber

Suppose we're given a fiber bundle over the circle, $\phi: E \to S_1$. Pick a representative element $\gamma: [0, 1]$ for the generator of $\pi_1(S_1, 1)$, and define the homotopy $h: F_0 \times [0, ...
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On the Euler Class of a manifold

Let $M$ be an orientable, compact $n$ dimensional differentiable manifold and $e \in H^n(M, \mathbb{Z})$ the Euler class of the tangent bundle of $M$, defined via the Thom Isomorphism. Also, let $[M] ...
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Kernel of induced map for fiber bundle on $G_n(\mathbb{C}^\infty)$

So let $$F_n(\mathbb{C}^n)\overset{i}{\rightarrow}F_n(\mathbb{C}^\infty)\overset{p}{\rightarrow}G_n(\mathbb{C}^\infty)$$ Where $F_n$ are $n$-tuples of orthogonal vectors and $G_n$ is the ...
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Cohomology ring of flag variety

So I was concerned with one of Hatcher's remarks. When he is computing the cohomology ring of the Grasmannian he makes a short aside by saying that $H^\ast(F_n(\mathbb{C}^k);\mathbb{Z})$ is equal to ...
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Sectional category (Schwarz genus) of the Milnor join construction

Assume topological spaces to be normal and paracompact. Following the article: "The genus of a fiber space" by A. Schwarz, we call the sectional category (or Schwarz genus) of a locally trivial fiber ...
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Classification of $O(2)$-bundles in terms of characteristic classes.

It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by their first Chern class. I was wondering what was the characterization of $O(2)$-bundles in terms ...
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Chern class of Hopf fibration: $S^1 \hookrightarrow S^3 \xrightarrow{\ p \, } S^2$?

As definition, the first Chern class is an element in $H^2_{dR}(S^2)$, how can we represent it as an integer? If we change the orientation of $S^2$, then does the integer change sign?
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Geometric intuition for homotopy invariance of fiber bundles?

There's a nice result in algebraic topology saying that given a fiber bundle, its pullbacks along homotopic maps are isomorphic as bundles. Thinking of a bundle as a comb with the "teeth" as its ...
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Fiber bundles that can be turned into a fibration that is a fiber bundle.

Let me recall a standard construction. Up to homotopy equivalence, any map $f: X \to Y$ is a fibering. Take the special case where $X=E$ the total space of a fiber bundle, and $Y$=B, the basespace ...
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What is the structure group of Hopf map $\pi :S^{15}\rightarrow S^8$?

What is the structure group of Hopf map $\pi :S^{15}\rightarrow S^8$? I know that $\pi :S^{15}\rightarrow S^8$ is a fiber bundle with fiber $S^7$. However $S^7$ cannot be a Lie group. But every fiber ...
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Vector Bundles: Continuity of map between total space implies homeomorphism.

In Spivak's "A Comprehensive Introduction to Differential Geometry" Spivak defines a vector bundle as a tuple: $(E, \pi, B, \bigoplus, \bigodot),$ where $E$ is the total space, $B$ is the base space, ...
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Explicitly constructing the Total Metric on a line bundle

Suppose that I'm given a Riemmanian manifold $<B,g_B>$ and a real line bundle $\pi:E\rightarrow B$, such that each fiber above any $b\in B$ comes equipped with a Riemmanian metric $g_b$. Then ...
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Is there a nontrivial fiber or principal bundle over $S^3$?

Is there a nontrivial fiber or principal bundle over $S^3$?I know that, by a paper of Steenrod,see the link below, every sphere bundle on 3- sphere is trivial but what about arbitrary fiber ...
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Is the transfer homomorphism in cohomology surjective?

Let $p:\widetilde{X} \to X$ be an $n$-sheeted covering space. Consider a singular simplex $c:\triangle^k \to X$, because the simplex is simply-connected, there exist $n$ different lifts ...
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Induced map in cohomology of a covering [closed]

Is it true that if $p: E \to B$ is a $2$-fold covering, the map $p^*$ induced in cohomology is surjective?
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A fibre bundle of n-tuples of vectors

The question is: does the surjective, multilinear map $\pi: (\mathbb R^\infty)^n\to \Lambda^n(\mathbb R^\infty)$ defined by $(v_1,\cdots,v_n)\mapsto v_1\wedge \cdots \wedge v_n,$ define a fibre ...
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Cohomology of a classifying space

I would like some advice on the following problem: I have a topological group $G=\langle H,g\rangle$, where $H$ is a subgroup and $h$ is an element. Using a result of classifying spaces $Bi:BH \to ...