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 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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21 views

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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16 views

Induced map in cohomology of a covering [on hold]

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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34 views

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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13 views

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 ...
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36 views

Is the associated bundle construction a bifunctor?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and ...
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31 views

Differential geometry of projective bundles

Can someone give me a reference about projective bundles from a differential-geometric point of view? I am not very familiar with algebraic geometry. I would like, for example, some theory about when ...
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22 views

Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold

The symplectic blow-up of a compact symplectic manifold $(X,\omega)$ along a compact symplectically embedded submanifold $(M,\sigma)$ results in another compact manifold $(\tilde{X},\tilde{\omega})$ ...
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50 views

What is a sympelctic bundle

What is a symplectic bundle? Is it a fibre bundle or a vector bundle? I am hoping for a not-very-technical answer because I'm not familiar with bundles in general. Sorry for that. PS: This symplectic ...
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34 views

How is this “Stiefel manifold”-like fiber bundle called? How to characterize it?

Let $M$ be a smooth manifold of dimension $n$. Consider the space $V_k(M)$ of pairs $(x,\alpha)$ where $x \in M$ and $\alpha$ is a linear embedding $\mathbb{R}^k \hookrightarrow T_x M$ (or ...
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37 views

What is the mathematical understanding behind what physicists call a gauge fixing?

I'm learning fiber bundle from my poor physicist point of view. I understand that a gauge transformation (physicist language) corresponds to the transformation of the connections built from an ...
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28 views

Differentiability of Hopf map

Let $S^{2n+1}$ be the unit sphere considered as a subset of $\mathbb{C}^{n+1}$ and define an action of $S^1 \subset \mathbb{C}$ by usual (complex) scalar multiplication. The quotient space is the well ...
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65 views

Defining a differentiable structure by means of functions.

I am trying to understand the construction of principal bundles from Kobayashi and Nomizu, and the situation is the following. Let $M$ be a manifold, $\{ U_\alpha \}_{\alpha \in A}$ an open covering ...
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2answers
71 views

Reference Request for Fibre Bundle Theory from the Smooth Manifold Point of View

I am looking for a book, or a set of notes, which discusses some basic theory of fibre bundles. I am interested more in the geometric aspect (smooth manifolds) rather than topological aspect. I found ...
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29 views

Why can one discriminate between the trivial $S^1$ line bundle and the Möbius strip by knowing the fibre transformation group?

Both $S^1\times\mathbb R$ and the Möbius strip can be regarded as line bundles over $S^1$. I have read that one can reconstruct a fibre bundle by knowing its base space, its fibre and the bundle ...
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61 views

Make this example of a coordinate bundle more concrete

I came across the following example of a coordinate fibre in the book "An introduction to differential manifolds": The (open) Mobius strip with $B=S^1$, $F=\mathbb{R}$, $G=\mathbb{Z}/2$. Here we ...
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36 views

Classification of $G$-principal Bundle and classification of $G$-coverings: a bridge between the two?

I encountered the following sentence in an exercise (the context is irrelevant) Let $G\cong\langle s_1,s_2,\dots , s_g \mid R \rangle$ be a discrete one relator group. Consider the $G$-principal ...
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45 views

Equivalence between smooth and topological fiber bundles

All manifolds in this post are hausdorff and second-countable. Is it true that two smooth fiber bundles with same fiber, base manifold and structure group (that is a Lie group $G$ of diffeomorphisms ...
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25 views

Compatibility of two notions of orientability for bundles

I have come across two notions of orientability : Notion 1 : A smooth manifold $M$ of dimension $n$ is said to be orientable iff $\exists$ a nowhere vanishing smooth $n-$form. The other notion in ...
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15 views

Must the induced iso. on $G$ of an aut. of a $G$-principal fiber bundle be the identity map?

I suppose the following definition is standard. Given a principal $G$-bundle $P(M,G)$ ($G$ acting freely on $P$ on the right) over some manifold $M$, an automorphism $j$ of $P$ is simply a bundle map ...
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54 views

Smoothness of transition maps of fiber bundle

A smooth fiber bundle with fiber a smooth manifold without boundary $F$ and structure group $G$ a Lie group of diffeomorphisms of $F$ is a smooth surjective map $p:E\to M$ between manifolds, with a ...
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25 views

When is a fibration (canonically) a principal fibration over its group of automorphisms?

The question is inspired by the following observation: Let $p: X'\to X$ be a connected covering space where both spaces are suitably nice (say they are CW complexes), then $p: X' \to X$ is a principal ...
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30 views

Is the base of a disc bundle necessarily a strong deformation retract of the total space?

I am reading Algebraic Topology by E.H.Spanier and in the proof of the Thom-Gysin map for disc bundles (on page 260) he says that $p : E \to B $ is a deformation retraction. I do not understand how ...
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62 views

Semantics: 'determinant bundle', top exterior power of vector bundle

From what I can dig up, given a vector bundle $E\rightarrow X$, the determinant bundle associated to this is $\Lambda^{n}E\rightarrow X$, where $n$ is the rank of $E\rightarrow X$. Is this the same ...
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79 views

Why are the fibers of principal G-bundles homeomorphic to G?

I'm trying to get a grip on the modern geometric formulation of gauge theory, in particular connections on principal G-bundles. However, I am stuck right after the definition already: Virtually all ...
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44 views

Isomorphism between $Hom(E,F)$ and $E^*\otimes F$, E and F vector bundles.

I need some aid in finding the solution to the following problem: Let $E$, $F$ be vector bundles (finite rank, if needed) over a manifold $M$. Consider the st $Hom(E,F)$ of all bundle morphisms from ...
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39 views

Computing Stiefel Whitney classes

I am computing the cohomology of $BO(2) \times B0(3)$ and I would like to identify the Stiefiel Whitney classes of this space. For instance, I know $H^*(BO(2);\mathbb{Z}/2)\cong ...
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36 views

Exterior power of cotangent bundle of a cartesian product of manifolds

Let $M \times F$ be a product manifold. The identity $T^*(M \times F) = T^*(M) \times T^*(F)$ holds generally (I hope). What can be said about the exterior powers of it? $$\bigwedge^r T^*(M \times ...
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21 views

Additive cohomology of BO(k)

we know that $H^*(BO(n);\mathbb{Z}/2) \cong \mathbb{Z}/2[w_1,\cdots,w_n]$, where $w_i \in H^i(BO(n);\mathbb{Z}/2)$ is the $i$-th Stiefel Whitney class. Is it correct to say $\langle w_i \rangle = ...
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40 views

Is multiplication by a Stiefel-Whitney class an injective map?

I have a doubt: In cohomology, when you multiply by a Stiefel-Whitney class is it always an injective map? For example: is $$H^{j-1}(X)\xrightarrow{\smile\ w_1}H^{j}(X)$$ always injective? Thanks!
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42 views

How to prove this sum being a real number related to unitary matrices?

Based on my recent study on non-Abelian gauge theory in physics, I encounter an identity that should be correct physically but I don't know how to prove it mathematically. Consider a $n\times n$ ...
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63 views

When are mapping tori isomorphic as bundles over the circle?

Suppose $\Sigma$ is an orientable genus-$g$ surface (possibly with boundary). The mapping torus corresponding to an orientation-preserving diffeomorphism $\phi: \Sigma \to \Sigma$ is the quotient ...
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24 views

Example of fibre bundle is locally product but not globally

When I read the below picture ,I can't make a example for claim of red box.
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23 views

fiber bundle of classifying spaces

Let $H,K$ be a connected Lie groups and $K$ a closed subgroup of H. Consider the bundle $EH/K\longrightarrow EH/H$ whose fibre is $H/K$. I want to know if $BG\times EH/K\longrightarrow EH/H$ is a ...
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71 views

Isoclinic rotations in four dimensions

Given any collection of complementary, oriented (2D) planes in n-dimensional space, and an angle associated with each one, there is a unique rotation of the whole space which restricts to rotations in ...
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1answer
26 views

Is the total space of a principal bundle parallelizable?

Given a smooth $G$-principal bundle $P \to M$, is $P$ in general parallelizable as a manifold? That is, is the tangent bundle $TP \to P$ trivial? In the case of Klein geometries, and more generally, ...
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38 views

Understanding the Ehresmann connection

I am trying to understand the concept of an Ehresmann connection on a fibre bundle $B$. Am I correct in saying that the connection gives the decomposition of every vector in $TB$ into the sum of a ...
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49 views

Covariant derivative from an Ehresmann connection on a fibre bundle

Given an Ehresmann connection on a fibre bundle, is it possible to define a covariant derivative that measures the rate of change of a section of the fibre bundle as you move through the base ...
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57 views

If two total spaces are locally homeomorphic to their base spaces, when will a fibre preserving map between them be continuous?

Let $fp = p'g$, where $f$ is continuous, $p$ and $p'$ are local homeomorphisms. When will $g$ be continuous ? My reading says it is iff any local (continuous) section $s$ of $p$ over $U$ which is ...
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30 views

When a continuous map from a quotient space satisfies the sheaf condition?

In the category of topological spaces, let E -> E/R -> X, where R is an equivalence relation. Say E -> X satisfies the condition (*) iff every point in E has a open nbd such that it homeomorphic to ...
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25 views

What's the sufficient and necessary condition for an equivalence relation making the quotient of a sheaf not only a fibre space but also a sheaf?

Let $p : E \to X$ be a sheaf and $R$ be an equivalence relation in $E$ (a subset of $E^2$) such that $p = E \to E/R \to X$ Clearly, $E/R$ is a fibre space on $X$ with the quotient topology. $E/R \to ...
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74 views

Are all fibres of a trivial fibre space over X with fibre F canonically homeomorphic with F?

In the category of topological spaces, a fibre space (E, X, p) is a triple of a morphism (continuous map) p of E to X. If j is a monomrphism of x to X where x is a final object, the fibre of x is a ...
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41 views

Definition of Normal Bundle and little exercise

I need to show that, given a manifold $M$ and its diagonal $\Delta\subset M\times M$, we have $T\Delta\cong\mathcal{N}_{\Delta|M\times M}$, where $T\Delta$ is the tangent bundle and ...
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33 views

Smoothness of local trivialisation of a vector bundle

I'm building a vector bundle structure for the space $L_{alt}^k(TM) = \bigcup_{p \in M}L_{alt}^k(T_pM)$, the bundle of alternating k-multilinear maps in $T_pM$. $L_{alt}^k(T_pM)$ is a vector space ...
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29 views

Relation between Thom class and closed Poincaré dual

Let $S,M$ be oriented manifolds without boundary such that $i:S\to M$ embedding and $i(S)$ is closed in $M$. $$ \eta_{S|M}:=\frac{TM_{|S}}{TS} $$ is the normal bundle of $S$ in $M$. I want to find a ...
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77 views

the first chern class of complex vector bundles

Let $\xi^\mathbb{C}$ be an $n$-dimensional complex vector bundle over a manifold $M$, $n\geq 2$. Question 1: Are there any practical methods to detect whether the first Chern class (with integer ...
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43 views

Is the unit bundle of a Finsler vector bundle a sphere bundle?

Note: By now I have asked this question also at mathoverflow. Let $E$ be a Finsler vector bundle* of rank $k$ over a manifold $M$. Does the unit "bundle" $UE$ admits a structure of a sphere bundle? ...
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43 views

The Normal Bundle of a level set is trivial

Let $M,N$ be manifolds and $f:N\to M$ an embedding. For every $p\in N$, $df_p:T_pN\to T_pM$ shows that $T_pN\subset T_pM$. For every $p\in N$, define the normal space of $N$ in $M$ at $p$ as $$ ...
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31 views

Is a fiber bundle specified by its fiber and its base space?

I want to model certain physical concept using fiber bundles, because I believe it is the most suitable language therefor. I know what both the base space and the fiber in this situation are. Do these ...
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18 views

Chart description of disc bundle over S^2

What is an explicit chart description of the bundle of $2$-discs in the cotangent bundle of the $2$-sphere? I mean the set $\{v\in T^\ast S^2\mid \|v\|\leq 1 \}$.