Tagged Questions

3
votes
0answers
42 views

What is the geometric meaning of powers of the first Stiefel-Whitney class?

If $M$ is an $n$-dimensional manifold, I know $\\w_1^n(M)$ is a Stiefel-Whitney number of the manifold, but what does its vanishing geometrically mean? More generally, does the Stiefel-Whitney height ...
6
votes
0answers
82 views

Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
6
votes
1answer
72 views

How to classify principal bundles over a 2 dimensional surface?

I just want to how much people know about this at the moment? I thought this is elementary and may be of execrise level, but a quick google search showed serious papers written on this subject (like ...
6
votes
0answers
114 views

How did Chern pictured the first Chern number?

The first Chern number $\cal C$ is known to be related to various physical objects. Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to ...
0
votes
1answer
63 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 $B$ ...
1
vote
0answers
73 views

Universal cover as a principal $\pi_1$ bundle.

Let $M$ be a connected manifold with universal cover $\tilde M$ and fix $x_0 \in M$. Then it is well-known that $\tilde M \to M$ is a principal $\pi_1(M,x_0)$ bundle. I'm a bit confused about the ...
2
votes
1answer
132 views

Trivialisation of the normal bundle of $S^1$

I'd like to show that the normal bundle of $S^1$ is trivial. That is, I want to find a homeomorphism $\varphi : NS^1 \to S^1 \times \mathbb R$ such that $\varphi \mid_{N_s S^1}$ is linear for every $s ...
2
votes
3answers
375 views

Understanding the trivialisation of a normal bundle

I've been looking for a definition of "trivialisation of normal bundle". I think I understand what a vector bundle, fibre bundle and a local trivialisation of either is. I also know what a tangent ...
6
votes
2answers
205 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 ...
3
votes
1answer
87 views

Torsion in $H^2(X,\mathbb{Z})$ induced by torsion in $H_1(X,\mathbb{Z})$

Let $X$ be a topological space. The universal coefficient theorem says that there is a short exact sequence $$ 0\rightarrow Ext(H_{k-1}(X,\mathbb{Z}),\mathbb{Z})\rightarrow ...
4
votes
0answers
105 views

Cohomology of fiber bundle with a section

Let $f:E\rightarrow B$ be a $C^{\infty}$-fiber bundle. Assume that there is a section $s:B\rightarrow E$ of this bundle. One easy consequence of the existence of section is that map $$ ...
2
votes
1answer
120 views

Different Euclidean metrics on a vector bundle

Suppose I have two Euclidean metrics $\mu_1, \mu_2$ on a given vector bundle $\xi$. Does anyone know of necessary and/or sufficient conditions to ensure that there is a homeomorphism $\phi: E(\xi) \to ...
12
votes
1answer
183 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 ...
6
votes
1answer
127 views

A bundle which is not associated to a vector bundle.

Let $F\rightarrow E\stackrel{\pi}{\rightarrow} B$ be a fiber bundle with structure group $G$. We know that if we can reduce the structure group to a subgroup of $GL_n$ for some $n$ ...
3
votes
2answers
135 views

Intuition for induced bundle.

If $X$ is a manifold, $G$ a compact Lie group, $E$ a principal $G$-bundle over $X$ and $V$ be a vector space on which $G$ acts. Then one can form the vector bundle $E \times_{G} V$ over $X$. What is ...
3
votes
1answer
76 views

String and BString

In one of the talks of J.P. May he mentioned some examples of structure groups and their classification spaces (he mentioned: O, U, SO, SU, Sp, Spin, String, Top, STop, F and SF). Most of them are ...
3
votes
1answer
139 views

$G/H$ with $H$ contractible a fibre bundle?

Suppose $G$ is a topological group and $H\leq G$ a normal/closed subgroup of $G$. If $H$ is contractible, does the quotient map $p: G\rightarrow G/H$ form a fibre bundle? Is there a more general ...
1
vote
1answer
148 views

associated bundle

Given a principal G-Bundle $P\rightarrow X$ and if we let $G$ act on itself by multiplication (denote this action by $\rho$) we obtain an associated bundle $P\times_{\rho} G=(P\times G)/\sim$ where ...
1
vote
0answers
99 views

How is a section viewed as a bundle map?

Suppose $\pi$ is the projection, $E$ total space, $B$ the base space, and $F$ the fiber. A section of a fiber bundle is a continuous map $f\colon B \to E$ such that $\pi(f(x))=x$ for all $x \in ...
1
vote
1answer
122 views

Example of a nontrivial fiber bundle with total space compact, spin, and $p_1=0$

I would really appreciate if anyone could provide me with an example of a locally trivial, but globally nontrivial, fiber bundle $Y\hookrightarrow Z \rightarrow X$, where $X$, $Y$, and $Z$ are all ...
2
votes
1answer
209 views

Cohomology rings of (some) sphere bundles over spheres

Recall that 2-dimensional complex vector bundles over $S^4$ are classified by $\pi_4(BU(2))=\pi_4(BU)=\mathbb Z$. For any integer $\lambda$ one can consider projectivisation of the corresponding ...
4
votes
0answers
125 views

Lie group quotient bundle with image of normalizer as structure group

In Glen Bredon "Topology and Geometry", Ch. II-13, I am stuck on the following "Problem 1. Finish the proof at the end of this section that $G\rightarrow G/H$ is a bundle. Also show that this is a ...
7
votes
1answer
143 views

Sanity check: Is every T-principal bundle over T trivial?

Is the following reasoning correct? The classifying space of the 1-torus $\mathbb T$ is $\mathbb{CP^\infty}$. Hence isomorphism classes of $\mathbb T$-principal bundles over $\mathbb T$ are in ...
2
votes
1answer
189 views

Octonionic Hopf fibration and $\mathbb HP^3$

Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations ...
2
votes
2answers
201 views

Classification of general fibre bundles

For principal $G$-bundles with $G$ a Lie group there exists a principal $G$-bundle $EG \to BG$ such that we have a bijection $$ [X,BG] \leftrightarrow \text{(principal $G$-bundles over X)} $$ $$ f ...
7
votes
1answer
237 views

Local triviality of principal bundles

Suppose I define a principal $G$-bundle as a map $\pi: P \to M$ with a smooth right action of $G$ on $P$ that acts freely and transitively on the fibers of $\pi$. Does it follow that $P$ is locally ...