3
votes
1answer
104 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
151 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 ...
2
votes
1answer
66 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 ...
2
votes
2answers
81 views

Is the following situation about first Chern numbers possible?

Let's consider complex vector bundles on a torus $T^2$ constructed in the following way: Suppose we have a map $f:T^2\to U(n)$, where $U(n)$ is the space of $n\times n$ unitary matrices. This ...
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 ...
2
votes
1answer
46 views

Stiefel-Whitney Classes: Simple Example

I need help finding the Stiefel-Whitney classes $w_k(\eta)$ of the normal bundle of the $n$-sphere. Now since $H^k (S^n ; \mathbb{Z}/2\mathbb{Z}) =0$ for $k \neq 0,n$, then $w_k(\eta) =0$ for $k ...
0
votes
3answers
51 views

Vanishing of non top-order Chern classes

Let $E \to B$ be a rank-$r$ complex vector bundle and denote by $c_1(E)$, $\ldots$, $c_r(E)$ its Chern classes. Then $c_r(E)$ is just the Euler class of the realization of $E$ as a real vector 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 ...
2
votes
0answers
84 views

How can I get 3264 conics from chern class?

I'm studying Algebraic geometry by "Enumerative Geometry And String Theory" [Katz]. In section 8.3, he computed the excess contribution 31 and concluded that the number of smooth conics tangent to ...
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 ...
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 ...
1
vote
2answers
143 views

Euler class of dual bundle

Let $L^*$ be the dual bundle of complex line bundle $L$. Since the bundle $L\otimes L^* = \text{Hom}(L,L^*)$ has nowhere vanishing section given by the identity map, the first Chern class ...
6
votes
1answer
216 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
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 ...
2
votes
0answers
131 views

Chern class of tautological line bundle

I'm studying characteristic classes from the Chern-Weil construction (via connection and curvature). I'm trying to compute some simple examples. Let $E$ be the tautological line bundle over projective ...
1
vote
0answers
44 views

Non-(stable)-triviality of the tautological bundles

The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann manifold $G_k(\mathbb{K}^N)$ of all $k$-planes in $\mathbb{K}^N$ (for $\mathbb{K} = \mathbb{R}$, $\mathbb{C}$ or ...
4
votes
1answer
407 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 ...
11
votes
1answer
361 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 ...
1
vote
1answer
141 views

Existence of a square root of a given line bundle via Chern class?

I come across a statement like Let $L$ be a complex line bundle on a manifold. $c_1(L)=0$ mod $2$ if and only if there exists a line bundle $K$ such that $L\cong K^{\otimes 2}$. How can one ...
2
votes
0answers
67 views

Is Transversality invariant by losing Dimension?

Setup Let $X$ be a smooth, reduced and irreducible scheme. And let $E$ be a vector bundle of rank $n$ on $X$. Following Eisenbud and Harris we say that the collection $\{\sigma_1,...,\sigma_k\}$ of ...
2
votes
0answers
51 views

Possible restriction on first nonzero Stiefel-Whitney classes?

I've been reading Hatcher's book on vector bundles and I'm just getting into the section of Steifel-Whitney numbers. Naturally, I'm interested in which sequences of such numbers are realizable, and ...
3
votes
1answer
159 views

Pontryagin classes of a product manifold

I'm imagining there's a way to relate the pontryagin classes of $T(M\times N)$ to the pontryagin classes of $M$ and those of $N$, but I haven't been able to find a helpful reference. Could someone ...
1
vote
1answer
630 views

Vanishing of Chern classes

I finally got the characterization of Chern classes, but i have another question: Let E be a Complex vector bundle of rank r on M. Given $s_1, \cdots ,s_r$ generic global sections, i can characterize ...
1
vote
0answers
308 views

Characterization of Chern classes and Whitney product formula

Let E be a Complex vector bundle of rank r on M. Given $s_1, \cdots , s_r$ generic global sections, i can characterize the i-th Chern class as follows: $ C_i(E)= \eta_{V_i}$, and $V_i$ is the locus ...
6
votes
1answer
182 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 ...
5
votes
3answers
1k views

Chern Classes of a Trivial Bundle

Could someone explain to me why the chern classes of a trivial bundle are zero? (I'm studying it from Bott & Tu book) To be more specific I can't understand why, given the vector bundle $E$ on ...