3
votes
2answers
122 views

What is the suspension used in the Freudenthal suspension theorem?

The theorem states: The suspension map $\pi_{i}( S^{n})\rightarrow \pi_{i}(S^{n+1})$ is an isomorphism when $i<2n-1$ and a surjection when $i=2n-1$. In the case where $X$ is an ...
3
votes
1answer
67 views

Is the Hopf fibration the only fibration with total space $S^{3}$?

Let $S^{1}$ bundle over $S^{2}$ is homeomorphic (diffeomorphic) to $S^{3}$. Is the Chern class of the fibration 1, i.e. is the Hopf fibration up to isomorphism the only fibre bundle with total space ...
6
votes
3answers
103 views

let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
1
vote
2answers
88 views

tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
4
votes
0answers
46 views

Fiber Bundle of Manifolds

I want to show the following: Let $E \rightarrow B$ be a fiber bundle with fiber $F$. Show that if $B$ and $F$ are manifolds, then so is $E$. Solving this problem seems easy enough simply by ...
2
votes
1answer
51 views

vector bundles and their cross-sections

Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...
1
vote
0answers
18 views

Equivalence between fibre bundles

Let $F$ and $U$ be two topological spaces on which a topological group acts (by the left) and consider the product action $G\times(F\times U) \rightarrow F\times U$ defined by $g(f,u)= (gf,gu)$. Now ...
1
vote
0answers
73 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
3
votes
0answers
41 views

how to obtain a generalized Morse function out of a fiber bundle?

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth ...
0
votes
0answers
53 views

Will the pullback of homotopic maps give rise to isomorphic fibre bundles?

I know it's certainly right for the case of vector bundles, but what about fibre bundles?
0
votes
1answer
43 views

Does the structure group of S^n homotopic to O(n+1)?

It is easy to show that Diff(S^1) is homotopic to O(2), but in the case of n bigger than 1 things become really complicated, I cannot see the conclusion directly.
1
vote
0answers
65 views

What's wrong with my argument to compute the vector bundles on $S^1$?

I have read the solution in Vector bundles of rank $k$ with base $S_{1}$, and I am sure that answer is correct. But I have another approach and I don't know where did I make a mistake. So as circle is ...
0
votes
0answers
54 views

Fibre homotopy equivalence

$E \mathop \to \limits^p B$ and $E_1 \mathop \to \limits^{p_1} B$ are two fibrations and there is a map $f:E \rightarrow E_1$ such that $f$ is a homotopy equivalence and ${p_1} \circ f = p$. Are ...
2
votes
0answers
59 views

Local coefficient System and the action of fundamental group

A local coefficient system $A\hookrightarrow E \to B$ is a fiber bundle $p:E\to B$ such that The fiber is a discrete abelian group $A$ The structure group $G$ is a subset of Aut$(A)$ Is the action ...
13
votes
1answer
219 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
4
votes
1answer
59 views

Local triviality condition on line bundles

We recall that a complex line bundle consists of a triple $(\pi,E,B)$ where $E,B$ are topological spaces, $\pi : E \to B$ a continuous map satisfying the following local triviality condition: ...
2
votes
0answers
86 views

Prerequisites for 'Fibre Bundles' by 'Dale Husemoller'

I wish to study the book 'Fibre Bundles' by Dale Husemoller. How much Algebraic Topology is required for studying this book ? Would a knowledge of fundamental groups, covering spaces (say from second ...
2
votes
1answer
136 views

Euler class of tangent bundle of the sphere

I am working through Milnor's Characteristic classes and am currently working problems on the topic of oriented bundles and euler class. I am having trouble computing the euler class of the tangent ...
1
vote
1answer
91 views

Global Section for Hopf Fibration

I want to know the existence of global section of $\pi : M\rightarrow M/G$, where $M$ is a Riemannian manifold with $G$-action. For instance in case of $M=S^2$ and $G={\bf Z}_2$ there exists no ...
4
votes
0answers
145 views

Cohomology of the pullback of a fiber bundle over the torus (generalised Eilenberg-Moore spectral sequence?)

I've found myself in the following situation and the tools that have been suggested to me to calculate the necessary invariants don't quite seem up to the job (I may be wrong in this as I have very ...
0
votes
0answers
33 views

Path fibration over a connected manifold.

Let $M$ be a differentiable manifold. We can consider $P(M):=\{\gamma:[0,1] \to M\}$, so we have a natural projection on $M$ $$ P(M) \to M $$ $$ \gamma \mapsto \gamma(1) ,$$ in the fibre of this ...
7
votes
0answers
229 views

When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
0
votes
0answers
64 views

Reduction of stucture group of tangent space

We consider the exat sequence of groups $$SO(n)\rightarrow O(n) \rightarrow Z_2$$ where the first map is the inclusion and the second is the determinant and the induced sequence $$BSO(n)\rightarrow ...
1
vote
0answers
60 views

Principal $G$-bundles as pull back bundles.

Let $G$ be a compact Lie group and consider a $G$-universal bundle $\pi: EG \to BG $ where $BG$ is the classifying space for the goup $G$ and the bundle $\pi: EG \to BG $ is defined as the principal ...
5
votes
1answer
136 views

Local coefficient System and universal cover

We work with a topological space $B$ which is path-connected and locally path-connected. I have troubles writing down a formal proof for the following proposition: Prop: Any local coeeficient ...
2
votes
2answers
83 views

Universal property of universal bundles.

A classifying space for a group $G$ is a topological space $BG$ with a principle $G$-bundle $p : EG \to BG$ where $EG$ is contractile, so that $BG = EG/G$. A classifying space is universal in the ...
4
votes
1answer
88 views

Classifying map

Let $\xi=(E,p,B)$ a principal $G$-bundle and $\eta=(P,\pi,Q)$ a real vector bundle such that $\operatorname{rank}(\eta)=n$. We can consider a classificant space $BG$. What is the classifying map $f:X ...
3
votes
0answers
55 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 ...
14
votes
1answer
176 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
0answers
133 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
161 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
79 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
90 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 ...
3
votes
1answer
159 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 ...
3
votes
3answers
888 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 ...
7
votes
2answers
307 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 ...
4
votes
1answer
92 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 ...
5
votes
0answers
206 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
159 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 ...
13
votes
2answers
277 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 ...
7
votes
1answer
155 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 $n$ ...
4
votes
2answers
216 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
83 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
161 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 ...
2
votes
1answer
173 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
111 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
144 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
271 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
140 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
191 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 ...