0
votes
0answers
12 views

Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
1
vote
1answer
40 views

Existence of a neighbourhood of a compact set ( from james fibrewise topology)

I'm reading James' Fibrewise topology book and I'm trying to understand the proof of proposition 7.4 , it says: Let X be a proper G-space . Then X is fibrewise regular over X/G. Proof For any $x \in ...
3
votes
1answer
46 views

How to understand structure groups?

I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following: A bundle is a tuple $(E,B,\pi)$ where $E,B$ are ...
2
votes
1answer
68 views

Contractibility of the space of sections of a fiber bundle

Let $\pi: E \to M$ a fiber bundle and $\Gamma(M,E)$ the space of smooth sections of the bundle with topology induced by the Whitney topology on $C^{\infty}(M,E)$. Assume that each fiber is ...
2
votes
1answer
92 views

Trivialisation of Moebius strip

I've just started studying Advanced Geometry and I'm in trouble with a (stupid) exercise. It's about finding a trivialisation of the Moebius strip (I'll refer to it as $E $) viewed as a fibre bundle ...
1
vote
1answer
94 views

Proving that the map from a topological space to a fiber bundle is open.

There are two pieces to a question, the first: Let G be a group acting on a space $E$. Show that the quotient map $E \rightarrow E/G$ is open. The second: Let p : E → X be a fiber bundle. Show that ...
2
votes
1answer
66 views

Fibers and they being discrete space

When talking about fibers in topology and geometry, especially in fiber bundle, vector bundle and fibration, it is said that they are discrete spaces. But in some diagrams I see, they are often ...
3
votes
2answers
145 views

Do fiber and structure group determine the fiber bundle over a given space?

So, let's say $B$ is a honest topological space (path connected and locally simply-connected), and we are given two fiber bundles over $B$ $$ F_i \hookrightarrow E_i \to B, \qquad i=1,2 $$ with ...
7
votes
3answers
239 views

Fiber bundle is compact if base and fiber are

I want to show that the total space $E$ is compact if the fiber $F$ and the base space $B$ are compact. Let $\pi$ denote the fiber projection. Since every point in $B$ has an open neighborhood $U$ ...
1
vote
0answers
30 views

$L^2$ of a fiber bundle

For spaces $X$ and $Y$ with measures there is an isomorphism of Hilbert spaces $$L^2(X) \otimes L^2(Y) \to L^2(X\times Y), ~~ f\otimes g\mapsto \left((x,y)\mapsto f(x)g(y)\right).$$ Now suppose $E ...
9
votes
1answer
786 views

An intuitive vision of fiber bundles

In my mind it is clear the formal definition of a fiber bundle but I can not have a geometric image of it. Roughly speaking, given three topological spaces $X, B, F$ with a continuous surjection $\pi: ...
1
vote
1answer
82 views

A question about the definition of fibre bundle

The canonical definition of fibre bundle is the following: Let $B,X,F$ be three topological spaces and $\pi:X\rightarrow B$ a continuous surjective map; then $(X,F,B,\pi)$ is a fibre bundle on $B$ ...
2
votes
0answers
151 views

Fiber Bundle: Hairbrush

I am trying to understand the hairbrush example of a fiber bundle from the Wikipedia article on fiber bundles. If I am understanding this, in the hairbrush example E is the hairbrush, ie. all the ...
7
votes
2answers
337 views

Which spheres are fiber bundles?

The Hopf fibration is a fiber bundle with total space $S^3$, and there are similar constructions for $S^7$ and $S^{15}$. Are there any other ways to regard a sphere as a nontrivial fiber bundle? My ...
6
votes
1answer
98 views

Restrictions for Principal Bundles on Manifolds

I have some manifold $M$ and am wondering what kind of Principal Bundles I am allowed to construct on it. To be more precise, what are the restrictions when trying to construct principal Bundles ...
13
votes
2answers
288 views

Given a fiber bundle $F\to E\overset{\pi}{\to} B$ such that $F,B$ are compact, is $E$ necessarily compact?

Consider a (locally trivial) fiber bundle $F\to E\overset{\pi}{\to} B$, where $F$ is the fiber, $E$ the total space and $B$ the base space. If $F$ and $B$ are compact, must $E$ be compact? This ...
0
votes
1answer
157 views

Is a sub-bundle of a vector bundle a vector bundle?

Could anyone please help me with this question? (1) Let (E, p, B) be a vector bundle where E is the total space, B is the base, and p is the structure map, that is, p:E->B. Now suppose E' is a ...
3
votes
1answer
162 views

Fiber bundle M x M - diagonal

Under what conditions for a space $M$ does the projection map to the first factor $p: M \times M - \Delta \rightarrow M$ has the local triviality condition, i.e. is a fiber bundle? Where $\Delta$ ...
2
votes
1answer
304 views

Principal and fiber bundles as defined by Husemoller

In his book 'Fiber Bundles' Husemoller defines principal bundles and fiber bundles quite differently from how they are usually defined. Specifically: Definition: a right $G$-space $X$ is called ...
1
vote
2answers
847 views

What is 'an identification map'?

From Husemöller's 'Fiber Bundles' (slightly rephrased): Proposition: Consider a bundle $\xi: E \to B$, and a mapping $f: B' \to B$. Then for any $s \in \Gamma(\xi)$ there is a $\sigma: B' \to ...
16
votes
1answer
597 views

What is the spectrum of the commutative C*-algebra I have constructed here?

Let $B$ and $F$ be compact Hausdorff spaces. Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$. I think this induces a fiber ...