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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Cohomology of the Thom Space of a Vector Bundle

The Thom space $T(E)$ of a vector bundle $E \to B$ with metric is defined as $D(E)/S(E)$, where $D(E)$ denotes the disk bundle and $S(E)$ denotes the sphere bundle of $E \to B$. I've been trying to ...
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A morphism of principal bundles is an isomorphism.

Let $p_1:E_1\to B$ and $p_2:E_2\to B$ be two principal $G$-bundles. Let $f:E_1\to E_2$ be a principal $G$-bundle morphism. I want to show that $f$ is an isomorphism. I read somewhere that it is ...
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Classify sphere bundles over a sphere

Problem (1) Classify all $S^1$ bundles over the base manifold $S^2$. (2) Do the same question for $S^2$ bundles. Moreover, does there exist a universal method to solve this kind of ...
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The pullback bundle

Consider a smooth fibre bundle $P:E\to B$ over a base $B$ with fiber $F$, and a trivializing family $\{(\phi_a:P^{-1}(U_a)\tilde =U_a\times F\}_{a\in A}$. Let the clutch functions of this family be ...
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Riemann surfaces with Riemann Roch theorem, linear fiber over an elliptic curve

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha z)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
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A differentiable fiber bundle exists for any family of clutching functions.

Lemma: Let $F$ be a smooth manifold and let $\{U_a\}_{a\in A}$ be a covering of a manifold $B$, and let $\{g_{ab}\}_{a,b\in A}$ be a family of clutching functions. That is, $g_{ab}:U_a\cap U_b\to ...
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The inverse of a smooth bundle map is smooth.

Consider smooth fiber bundles $P_i:E_i\to B_i,\quad i=1,2\quad$ with fiber $F$. Let $\tilde f:E_1\to E_2$ be a smooth bundle map. That is a smooth map which preserves the fiber $F$. Let $f:B_1\to B_2$ ...
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Pull-back of a fibration along a homotopy equivalence

Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\rightarrow B$ and $g:B\rightarrow B'$ be homotopy inverses. Denote by ...
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What is an infinitesimal automorphism of a connection?

Let $P\to M$ be a principal bundle over a differentiable manifold $M$, with fibre $G$, equipped with a connection $H\subset TP$. I have heard the term "infinitesimal automorphism of $H$" but I haven't ...
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Holomorphic Hermitian metrics

Let $E\to M$ be a complex vector bundle. A hermitian metric $h$ on $E$ is a hermitian inner product on each fiber $E_{p},\, p\in M$. Suppose that $M$ is also a complex manifold and that $E$ is ...
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How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible ...
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What the curvature $2$-form really represents?

Let $(E,\pi,B)$ be a principal bundle with structure group $G$. The connection $1$-form can be thought of as a projection on the vertical part. It allows us to characterize the horizontal subspaces as ...
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Fibre bundles for homogeneous spaces

Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. Then it is known that $G/H$ can be equipped with a unique differentiable structure such that $G\xrightarrow{\pi} G/H$ is a locally ...
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55 views

Euler characteristic of a singular fiber

I am trying to understand Kodaira's classification of fibers. In the table at page 41 of Miranda's book http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf there is given the Euler number of the ...
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Fiber bundle beginner question.

I'm reading some notes on fiber bundles. Let $f:X \rightarrow Y$ be a continuous map of topological spaces. The author states: We say $f$ makes $Y$ a fiber space over $X$ if $f$ is locally trivial ...
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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!
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When the associated bundle to a $U(1)$-bundle is an holomorphic line bundle?

Let $P$ be a principal $U(1)$ bundle over a complex manifold $M$, and let $\rho\colon U(1)\to Aut(\mathbb{C})$ be the representation of $U(1)$ on $\mathbb{C}$ given by the standard multiplication. My ...
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Correspondence between line bundles and $U(1)$-bundles: a mistake from the physicists?

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic line bundle equipped with a Hermitian metric $h$ and Chern connection $\nabla$. If ...
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Reduction of the structure group

Given a vector bundle $V$ of rank $r$ over a curve. What proprieties should $V$ satisfies in order to admet a reduction of the structur group to $O_r(k)$ (resp. $SL_r(k)$, $Sp(r,k)$) Could you give ...
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Is there a fibre bundle with fibre homeomorphic to $\mathbb R^k$ which cannot be given the structure of a vector bundle?

We define a (rank $k$) vector bundle to be a fibre bundle with fibre $\mathbb R^k$ such that the local trivializations are fibre-by-fibre vector space isomorphisms. I'm curious whether this linearity ...
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Metric, torsion free connections on principal bundles

Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold $M$, and let $H\subset TF(M)$ be a connection. A Riemannian metric on $M$ can be equivalently written as an equivariant ...
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Space of invariant sections

I have a principal bundle $\pi: M \to B$ with structure group $G$ and a vector bundle $p: V\to M$. I need to work with $\Gamma_G(V,M)$, the space of invariant (under the fiber action on $M$) sections ...
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Curvature of a principal bundle and the exterior covariant derivative

Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a connection on $P$, where $\mathfrak{g}$ is the Lie algebra of $G$. Associated to every ...
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Is restriction of a chart is a chart necessarey in the Definition of Fibre Bundle

This is the definition of Fibre Bundle from the notes James F Davis and Paul Kirk: I think the condition 3 is superfluous. Because if you have a chart over $U$ $\phi : U \times F \rightarrow ...
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Fibered product of Hopf Fibrations

I am wondering: what is the vector bundle associated to the Hopf-fibration $S^{3}\to S^{2}$? What is the fibered product of two Hopf-fibrations? Thanks.
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Fibered product of symplectic fiber bundles

Let $P_{1}\to B$ and $P_{2}\to B$ be smooth fiber bundles over the same base manifold $B$, and let us assume that the total spaces $(P_{1},\omega_{1})$ and $(P_{2},\omega_{2})$ are symplectic ...
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Morphism of vector bundles covering maps of the bases

Let $E_{1}\to B_{1}$ and $E_{2}\to B_{2}$ be vector bundles over differentiable manifolds $B_{1}$ and $B_{2}$. What means that $F\colon E_{1}\to E_{2}$ is a morphism of bundles covering a map $f\colon ...
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28 views

Enumerating fiber bundles with fiber and base first Eilenberg-Maclane spaces

How to enumerate fiber bundles (maybe, only as spaces, not bundles) with fiber $K(A, 1)$ and base $K(B, 1)$? It seems to be connected with enumerating short exact sequences of form $0 \to A \to ... ...
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Proof of $G\rightarrow G/H$ is a Principal H bundle

Let $G$ be a Lie group and let $H$ be a closed subgroup (not necessarily normal). Then $G$ is a principal $H$-bundle over the (left) coset space $G/H$. I could proof that the fibers are all ...
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61 views

Inner product on the space of sections

Let $L\to M$ be a real line bundle over a manifold $M$, and let us denote by $\Gamma(L)$ its space of sections. I am trying to find a product in $\Gamma(L)$ to make it into an algebra. The naive ...
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Size of largest fiber

Suppose I have a map $f\colon X\to Y$ with finite fibers (where $X,Y$ are simply topological spaces for now, although maybe it's better to think of schemes). Is there a name for the quantity ...
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Locally a product equals fiber bundle?

I know the definition of a fiber bundle as a map $p:E \rightarrow B$ such that the preimage of any open set in $B$ is diffeomorphic to a product and fits in a commutative diagram with the projection ...
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given a specific vector bundle how to see whether the first Pontryagin class is zero

Let $Z_2$ be the group with $2$ elements. Let $a\in Z_2$ be the nontrivial element. Let $S^n$ be the $n$-sphere. Let $C(S^n,2)=\{(x,y)\in S^n\times S^n\mid x\neq y\}$. Let $a$ act on $C(S^n,2)$ by ...
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Is there a fiber bundle approach to nonlinear oscillations?

I've recently been learning about nonlinear oscillations, and I noticed a seemingly strong connection between how the equations of motion are solved/approximated, and fiber bundles (or vector bundles ...
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Relationship between $\Gamma (E)\otimes \Gamma (B) $ and $\Gamma(E\otimes B) $

I have a question that I've been thinking about for a while. So If $(E,\pi_E)$ and $(B,\pi_B)$ are some vector bundles over some mainfold $M$. What is the exact relationship between $$\Gamma ...
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how to see whether a bundle is trivial or not?

Let $Z_2$ be the group with $2$ elements. Let $a\in Z_2$ be the nontrivial element. Let $S^n$ be the $n$-sphere. Let $C(S^n,2)=\{(x,y)\in S^n\times S^n\mid x\neq y\}$. Let $a$ act on $C(S^n,2)$ by ...
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principal bundle and the associated bundle

Let $G\leq O(n)$ be a subgroup of orthogonal group. Let $\xi$ be a principal $G$-bundle. Let $\xi[\mathbb{R}^n]$ be the associated vector bundle. If $\xi$ is not a trivial bundle, can we obtain that ...
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Projection of fiber bundle is a submersion

I'm just wondering about my proof for the following fact. I get the feeling it is almost trivial but I am still getting a feel for geometry and so it doesn't seem 'obvious' to me just yet. The ...
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Two different notions of covering homotopy?

In Steenrod's The Topology of Fibre Bundles, there are two different notions of covering homotopy. One is furnished in Theorem 11.3: Let $\mathcal B,\mathcal B'$ be two bundles having the same ...
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existence and uniqueness of projection maping

This is a question on Linear algebra by Greub. Let E be a vector space. Let $C(E)$ be the free vector space formed by E. Let $i_E:E\to C(E)$ be $i_E(x)=f_x$ where $f_x(y)=1$ if $y=x$ and $f_x(y)=0$ ...
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Terminology question : “half smooth, half topological” fibre bundle

First, I know (or I think I know...) the definition of fiber bundle, be it in the smooth or topological category. Here is my situation, which is kind of between the two: I have a smooth manifold $E$, ...
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Elementary proof of the fact that any orientable 3-manifold is parallelizable

A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors ...
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What is the topology of this quotient of $S^2 \times S^1$?

So suppose you take an $S^2$, then you put an $S^1$ fiber over it which degenerates by smoothly shrinking to a point at its poles. What is the topology of this space in more familiar terms (assuming ...
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Fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. ...
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line bundles over the circle

I read in various places that up to isomorphism there are only two line bundles ( 1-d vector bundles) over a circle, the trivial one and the mobius strip. On the other hand, when I make a mobius strip ...
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Fiber bundles and manifolds

Each vector bundle is an example of a fibre bundle with some extra structure. This extra structure provides the algebraic object consisting of all sections (continuos or smooth) of given bundle. When ...
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Does the classifying map of a fibre bundle only depend on the transition functions?

Does the classifying map of a fibre bundle only depend on the transition functions? Precisely, Let $\xi$ and $\eta$ be two fibre bundles over $B$, whose transition functions are same, both of their ...
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What can we say about a vector bundle with trivial unit sphere bundle?

If you want to avoid the back story to this question, feel free to skip the paragraphs between the horizontal lines. Yesterday Georges Elencwajg asked me the following question (I'm paraphrasing): ...
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Trivializations of Möbius band

If $E=\mathbb{R}\times (-1,1)$ and $\forall (x,t), (y,t^{'}) \in \mathbb{R}\times (-1,1)$, $$(x,t)\backsimeq(y,t^{'})\Leftrightarrow \exists n\in \mathbb{Z} ; x=y+2n\pi , t=(-1)^{n} t^{'} $$ ...
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Spin structures, frame bundles, and trivializations over the 2-skeleton

While reading an introduction to Spin- and Spin$^{\operatorname{c}}$ structures (found here), I encountered the following definition: Let $E\to X$ be an oriented $\mathbb{R}^n$-bundle over a CW ...