Tagged Questions

4
votes
1answer
57 views

Computing Chern Classes of Tautological Line Bundles

I cannot find any references with how to handle this tautological line bundle $V \rightarrow P(\mathbb{H}^2)$ where $\mathbb{H}$ are the Hamilton quaternions. I know it is a complex rank 2 vector ...
3
votes
0answers
62 views

How to calculate characteristic classes of tensor products?

I was given the following as an execrise: Prove that the tensor product of complex line bundles over $X$ satisfies the following relationship: $$c_{1}(\otimes E_{i})=\sum c_{1}(E_{i})$$ It is ...
1
vote
0answers
41 views

Canonical bundle and Möbius bundle

I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Möbius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
0
votes
0answers
17 views

Isomorphism canonical and Moebius bundle.

I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Moebius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
0
votes
1answer
64 views

Universal bundles and classificant maps.

We have this theorem: Let $\xi=(E,p,X)$ be a vector fiber bundle with rank($\xi)=n$. We can find a map $f: X \rightarrow Gr_n(\mathbb{C}^{\infty})$ such that $\xi=f^*(\gamma_n)$. I denote with ...
6
votes
3answers
59 views

introductory reference for Hopf Fibrations

I am looking for a good introductory treatment of Hopf Fibrations and I am wondering whether there is a popular, well regarded, accessible book. ( I should probably say that I am just starting to ...
2
votes
1answer
41 views

Classification of flat complex line bundles

I'm having a contradiction with two different classifications of flat complex line bundles over a manifold $X$. Suppose for simplicity that $H^2(X;\mathbb Z) = 0 = H^1(X;\mathbb C)$. Then the only ...
5
votes
1answer
120 views

Second Stiefel-Whitney Class of a 3 Manifold

This is exercise 12.4 in Characteristic Classes by Milnor and Stasheff. The essential content of the exercise is to show that $w_2(TM)=0$, where $M$ is a closed, oriented 3-manifold, $TM$ its tangent ...
2
votes
0answers
42 views

extending trivializations of plane bundles via obstruction theory

I came across the following statement: A trivialized $k$-plane subbundle of a trivialized $(n+k)$-plane bundle determines a trivialization of the orthogonal $n$-plane bundle over the $(n-1)$-skeleton ...
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 ...
3
votes
1answer
157 views

Classify the vector bundles of a manifold.

I met a question asking me to classify the $2$-dimensional vector bundles of the sphere $S^2$. I did not know how to classify the vector bundles in general. The only example I know was the line ...
4
votes
0answers
90 views

Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?

Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that ...
1
vote
1answer
61 views

Is the total space of this vector bundle embeddable into $\mathbb{R}^3$?

Let $M$ be the Moebius vector bundle over $S^1$. Is it possible to embedd the total space of $M\oplus (S^1\times \mathbb{R^1})$ over $S^1$ into $\mathbb{R}^3$? I suppose this isn't possible but I ...
4
votes
0answers
70 views

Decomposition of vector bundles over a CW complex

Let $X$ be CW complex having only cells up to dimension $n$. I want to prove that every $m$-dimensional vector bundle $E$ over $X$ decomposes as a sum $E\cong A\oplus B$ where $A$ is a ...
2
votes
1answer
110 views

Example of non-isomorphic vector bundles with the same element in $K$

Let $X$ be a paracompact and well behaved space. Topological K-Theory $K(X)$ of $X$ is group completing the monoid of isomorphism classes of vector bundles over $X$ with the Whitney sum. Two vector ...
6
votes
1answer
150 views

obstruction cocycle of stiefel manifold

I was reading about the interpretation of Stiefel Whitney classes as obstructions from Milnor-Stasheff's book and I got stuck at a step. The context is the following. Let $E \to B$ be a vector bundle ...
6
votes
3answers
150 views

Is there a way to establish a correspondence between the vector bundles over a torus and some kind of homotopy groups, just as we do to spheres

By introducing the "clutching function" one can relate the complex (real) vector bundles on a sphere with homotopy groups of $GL_n(\Bbb C)$ ($GL_n^+(\Bbb R)$ for oriented ones). Can we do a similar ...
5
votes
1answer
200 views

On Frobenius reciprocity theorem

The classical Frobenius reciprocity theorem asserts the following: If $W$ is a representation of $H$, and $U$ a representation of $G$, then $$(\chi_{Ind W},\chi_{U})_{G}=(\chi_{W},\chi_{Res ...
1
vote
1answer
187 views

If bundle map is a local isomorphism, then bundle map is diffeomorphic?

Suppose $\phi$ is a bundle map from $E$ to $F$, where $E$ and $F$ are two vector bundles over a manifold $M$. Now for each $x\in M$, $\phi$ restricted to $E_x:E_x\to F_x$ is a isomorphism, how to show ...