0
votes
1answer
19 views

Is the tangent bundle of an oriented surface plus a trivial bundle trivial?

Let $\Sigma$ be an oriented closed surface and $E$ be the direct sum of $T\Sigma$ with a trivial line bundle. Is $E$ a trivial rank $3$ vector bundle? For genus $0$ and $1$ the answer is yes since ...
4
votes
1answer
90 views

Working out an example in Hatcher vol. $2$: Pullbacks of the Möbius Bundle

I'm working out the examples made by Hatcher to shows some pullbacks (definition here for clarity) and this ("simplified" version with $n=2$ or $n=3$) gave me an hard time: $$\times \times \times ...
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
69 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
5
votes
1answer
96 views

Visualizing topology of a Vector Bundle

I've started reading Milnor, Stasheff - Characteristic Classes and at page $18$ they proved that $\mathbb{R}^n$-bundle $\xi$ is trivial if and only if $\xi$ admits $n$ cross sections $s_1, \dots , ...
3
votes
0answers
31 views

Total space of a geometric vector bundle

Suppose that $X$ is a scheme with a $B$-action, and that $\lambda: B \to \mathbb{C}^*$ is a character of $B$. We then have a geometric vector bundle $\pi: X \times^B k \to X/B$, where $B$ acts on $X ...
1
vote
0answers
39 views

Restriction of a line bundle to a a two-cycle

I am reading a paper on Chiral Differential Operators http://arxiv.org/pdf/hep-th/0604179v3.pdf and it says on page 23 that a line bundle over a manifold $C$ can be characterized completely by its ...
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 ...
1
vote
0answers
54 views

Relative Euler class

In this topic http://mathoverflow.net/questions/93438/euler-class-in-the-non-compact-case you can read about relative Euler class. Can you show me some example of calculation of this class? Do you ...
4
votes
2answers
90 views

The complex tangent bundle of $\mathbb{C}P^1$ is not isomorphic to its conjugate bundle.

In " Characteristic Classes" by Milnor and Stasheff on pages 167-168, the authors give a brief argument about why: The complex tangent bundle of $\mathbb{C}P^1$ is not isomorphic to its conjugate ...
0
votes
1answer
98 views

Stiefel classes and generic sections

One can say that the Stiefel-Whitney classes is dual classes to the locus of linearly dependence of generic sections. What means "generic"? It would be great, if you show me some relation in local ...
3
votes
0answers
75 views

Almost complex structures on a manifold and its exotic copy

Consider a compact simply-connected smooth $4$-manifold $M$ and its exotic copy $M'$. Identify the underlying topological manifolds. Dold-Whitney theorem guaranties ($\mathbb R$-linear) isomorphism of ...
6
votes
0answers
115 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
2
votes
1answer
68 views

self-intersection of lagrangian submanifold

Let's consider lagrangian submanifold $X$ in symplectic manifold $M$. Is it true that self-intersection index of $X$ is equal to the Euler characteristic $\chi(X)$? Can we construct (not canonical) ...
4
votes
2answers
152 views

Normal bundle in tangent bundle

Let's consider the normal bundle $NM$ of zero section in $TM$. Is it true that $NM \cong TM$? There is exact sequence $$0 \rightarrow TM \rightarrow TE|_M \rightarrow NM \rightarrow 0$$ for the ...
6
votes
1answer
192 views

To prove a vector bundle $\xi$ is orientable $\iff w_1(\xi)=0$

I went about it this way: $\xi$ an $n$-rank vector bundle over a topological space $M$ is orientable. $\iff$ its top exterior power $\bigwedge^n\xi$ is trivial $\iff$ the 1-frame bundle ...
3
votes
0answers
71 views

How should I imagine the cup product on a topological space/manifold/variety

Let $X$ be a "space" with a vector bundle $E$. Let $n$ be the dimension of $X$. Then there is a map $$H^1(X,E)\otimes \ldots\otimes H^1(X,E)\to H^n(X,\Lambda^n E).$$ This map is given by taken the ...
7
votes
2answers
134 views

Are there any simply connected parallelizable 4-manifolds?

On pp. 166 of Scorpan's "The Wild World of 4--manifolds", he gives an example of a parallelizable 4-manifold ($S^1\times S^3$) and then asserts: "there are no simply connected examples". It's ...
3
votes
1answer
80 views

Dropping the orientable condition from the Thom isomorphism theorem.

My first question is: what are some examples of un-oriented n-plane bundles over $ \mathbb{Z}$ where it is easy to see that there is no Thom class? I would like to know some examples because the real ...
4
votes
1answer
160 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 ...
4
votes
1answer
261 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
60 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
1answer
31 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
72 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
118 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
114 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 ...
8
votes
1answer
269 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 ...
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 ...
4
votes
1answer
321 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
121 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
64 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
85 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
149 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
171 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
182 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
369 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
201 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 ...