0
votes
1answer
18 views

Complex bundles on $S^{2n+1}$

We know that the complex $K$ theory of spheres are 2 periodic. On the other hand every complex bundle on $S^{1}$ is trivial. So $K(S^{2n+1})=0$. So this is a motivation to ask: Is there a ...
1
vote
1answer
46 views

Real vector bundles on $S^{7}$

Is it true that $\pi_{6}(O(n))=0$ for all n? Equivalently, are all real bundles on $S^{7}$ trivial?
2
votes
0answers
35 views

Universal property of tensor product of vector bundles

To define the tensor product of vector bundles $\xi_1$ and $\xi_2$ over base $B$, Milnor-Stasheff's Characteristic Classes takes the space $\sqcup_{b \in B} F_b(\xi_1) \otimes F_b(\xi_2)$ and ...
1
vote
1answer
28 views

spin structure definition

Suppose we have a principal $SO(n)$-bundle $E$ over $B$, with projection map $p$. We say that it admits a spin structure if there is a prinicipal $spin(n)$-bundle $E'$ over B, with projection map ...
2
votes
1answer
35 views

K-theory product trivial on $K^1$ due to stability?

Let $K^0$ be the topological complex vector bundle $K$-theory functor. On the one hand, we have that the product on $K^0(\Sigma X)$ vanishes by a Mayer-Vietoris argument. On the other hand, we define ...
0
votes
0answers
13 views

Vector bundle, nonexistence of Euclidean metric

Milnor-Stasheff "Characteristic classes" problem 2-C says: Any vector bundle over a paracompact base space can be given a Euclidean metric in other words, if $\pi : E \rightarrow B$ is a vector ...
1
vote
1answer
108 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
3
votes
1answer
102 views

Characteristic Class with arbitrary coefficient

I'm looking for some "natural definition" for characteristic class with arbitrary coefficient. For example, the chern class satisfies four conditions and Stiefel-Whitney class satisfies three ...
6
votes
1answer
142 views

Tensor product of real line bundles is trivial as a map $\mathbb{R}P^\infty\to\mathbb{R}P^\infty\times\mathbb{R}P^\infty\to\mathbb{R}P^\infty$

The tensor product of a real line bundle with itself is trivial, as is easily seen by looking at the transition functions or checking the Stiefel-Whitney class. Real line bundles are classified by the ...
3
votes
1answer
45 views

The long exact sequence in reduced K-theory. How to glue together the short ones?

I'm trying to filling the details about the construction of the long exact sequence in K-theory. I'm using the notation of Hatcher's book (pages 52-53). here is a related question, even thought it's ...
2
votes
1answer
53 views

“Visual” interpretation of the Bott Periodicity for complex vector bundles

I've just read the Bott Periodicity proof (complex case) in Hatcher book about K-theory. At first it seems that using this theorem I could conclude that every complex vector bundles over $S^{2n+1}$ ...
2
votes
0answers
20 views

Euler class of quotient bundle of real projective space

Let $\gamma$ be the real tautological line bundle over the real projective space $\mathbb{R}P^n$, $V$ be the trivial bundle of rank $n+1$, and $Q$ be the quotient bundle $V/\gamma$. What is the Euler ...
2
votes
1answer
64 views

Characteristic classes not defined on vector bundles

If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking ...
4
votes
1answer
45 views

Characteristic classes for quaternionic bundles

In a nutshell, the derivation of the Stiefel-Whitney and Chern classes for a (let's say) closed space $X$ is as follows: For $k = {\mathbb{R}}$ or $\mathbb{C}$, any $k$-line bundle $\xi \to X$ is the ...
2
votes
0answers
71 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
2
votes
0answers
61 views

Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
2
votes
2answers
94 views

Definition of Normal Bundle

I'm reading Differential Forms in Algebraic Topology by Bott and Tu. I reached the point where the book defines the normal bundle of a submanifold and uses the tubular neighborhood theorem. I can't ...
2
votes
1answer
44 views

Total space of vector bundle deformation retracts onto 0-section of base space

I'm trying to prove the following: Total space of vector bundle deformation retracts onto 0-section of base space. Books seem to take this fact for granted. I checked Bott Tu and Hatcher. Online ...
0
votes
0answers
42 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
0
votes
0answers
66 views

Two books defined two Chern(Euler) classes yet differed by a negative sign, what's wrong?

In the book 'Principles of Algebraic Geometry' P141 and the book 'Differential Forms in Algebraic Topology' P72-73, they defined the Chern(Euler) classes of line bundles using the patching data ...
0
votes
1answer
23 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
105 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
58 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
77 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
137 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
34 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
43 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
84 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
65 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
102 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
113 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
87 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 ...
8
votes
2answers
226 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
83 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
205 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
215 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
73 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 ...
9
votes
2answers
150 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
101 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
207 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
403 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
65 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
37 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
76 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
131 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
132 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 ...
11
votes
1answer
360 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
180 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
1k 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 ...
5
votes
1answer
395 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 ...