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)

learn more… | top users | synonyms

2
votes
0answers
26 views

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 ...
0
votes
1answer
28 views

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 ...
1
vote
1answer
42 views

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 ...
8
votes
0answers
64 views

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 ...
6
votes
1answer
76 views

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 ...
1
vote
0answers
26 views

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 $A$ ...
0
votes
0answers
15 views

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 ...
1
vote
0answers
12 views

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 ...
1
vote
0answers
20 views

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 ...
1
vote
1answer
33 views

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?
1
vote
0answers
26 views

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 ...
0
votes
1answer
22 views

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 ...
-1
votes
2answers
65 views

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 ...
1
vote
0answers
45 views

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 ...
0
votes
0answers
10 views

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 ...
0
votes
1answer
16 views

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 ...
1
vote
0answers
24 views

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 ...
0
votes
1answer
32 views

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*} ...
0
votes
0answers
13 views

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 ...
2
votes
1answer
20 views

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 ...
0
votes
1answer
18 views

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 ...
0
votes
1answer
15 views

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 ...
0
votes
0answers
22 views

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, ...
1
vote
1answer
28 views

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] ...
0
votes
0answers
24 views

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 ...
2
votes
0answers
37 views

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 ...
1
vote
0answers
21 views

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 ...
3
votes
1answer
63 views

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 ...
0
votes
0answers
33 views

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?
3
votes
1answer
33 views

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 ...
2
votes
1answer
78 views

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 ...
2
votes
1answer
27 views

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 ...
3
votes
1answer
37 views

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, ...
1
vote
1answer
27 views

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 ...
2
votes
1answer
77 views

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 ...
1
vote
1answer
22 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 ...
0
votes
1answer
19 views

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?
1
vote
0answers
41 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 ...
1
vote
0answers
15 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 ...
1
vote
1answer
44 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 ...
1
vote
1answer
32 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 ...
0
votes
1answer
28 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 $(\tilde{X},\tilde{\omega})$ ...
1
vote
1answer
54 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 ...
2
votes
1answer
42 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 ...
5
votes
0answers
48 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 ...
0
votes
0answers
37 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 ...
0
votes
2answers
71 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 ...
2
votes
2answers
96 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 ...
2
votes
1answer
45 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 ...
3
votes
0answers
64 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 ...