Tagged Questions

3
votes
1answer
67 views

Does the splitting principle define chern classes for vector bundles if they are known for line bundles?

Suppose we have definied chern classes $c_i$ for line bundles for all $i\geq 0$. Let $E\to B$ be a rank $r$ vector bundle. By the splitting principle there exists a map $f:Y\to B$ such that $f^*E$ is ...
3
votes
0answers
42 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 ...
0
votes
1answer
28 views

Chern classes tangent bundle $\mathbb{C}P^n$.

Let $V \in Vect_k(M, \mathbb{C})$. We define Chern classes $c_i(V) \in H^{2i}(M, \mathbb{Z})$ with the usual 4 axioms. Now we consider the tangent bundle $$ \mathbb{C}^n \hookrightarrow T\mathbb{C}P^n ...
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 ...
2
votes
1answer
38 views

Curvature form, tangent bundle and structural group.

Let $T\mathbb{C}P^n$ the tangent bundle over $\mathbb{C}P^n$. We have that the Chern classes are the coefficients of the characteristic polynomial of curvature form $\Omega$ of $T\mathbb{C}P^n$: $$ ...
3
votes
2answers
95 views

An alternative description of the first Stiefel-Whitney class

I've heard this description of the first Stiefel-Whitney class of a vector bundle but I don't know why this is true. Can anyone help me with this, please? The first Stiefel-Whitney class of a vector ...
2
votes
0answers
46 views

Chern classes tangent bundle.

I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over $\mathbb{C}P^n$ ($T\mathbb{C}P^n$) and develop the theory beginning from this ...
5
votes
0answers
53 views

explicit cocycle representing Stiefel-Whitney class in Milnor and Stasheff

I am trying to do Problem 7A in Characteristic Classes by Milnor and Stasheff which asks the reader to do the following: Identify explicitly the cocycle $C^r(G_n) \cong H^r(G_n)$ which ...
4
votes
1answer
85 views

Stiefel-Whitney numbers for product bundle

I'm reading Milnor's "characteristic classes" and I want to compute Stiefel-Whitney numbers of $ P^2 \times P^2 $ (product of projective spaces) for one of the problems, I know how Stiefel-Whitney ...
5
votes
0answers
87 views

Exercise 4-A “Characteristic Classes” by Milnor and Stasheff

Exercise 4-A of Milnor and Stasheff's book "Characteristic Classes" reads: Show that the Stiefel-Whitney classes of a Cartesian product are given by $w_k(\xi\times\eta) = \sum^k_{i=0} ...
11
votes
2answers
136 views

Different ways of representing a second cohomology class

There are probably many ways of talking about a second (integral) cohomology class of a smooth, closed, orientable manifold $M$ of dimension $n$. Here are a few, with $\alpha\in H^2(M,\mathbb{Z})$: ...
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 ...
4
votes
0answers
143 views

Grothendieck on (topological) Chern Classes

I have been reading through the wikipedia article about Chern classes and it currently has a section devoted to the Alexander Grothendieck axiomatic approach. The language used throughout the section ...
2
votes
1answer
69 views

Number of Zeros of a Section vs Integral First Chern Class

I've often read that the first chern class can be seen as "the number of zeroes a section must have". How precise can this statement be made? I'm only interested in Line bundles. I actually know ...
6
votes
0answers
179 views

Cohomology ring of Grassmannians

I'm reading a paper called An Additive Basis for the Cohomology of Real Grassmannians, which begins by making the following claim (paraphrasing): Let $w=1+w_1+ \ldots + w_m$ be the total ...
4
votes
2answers
258 views

The “Wu formula” and Steenrod algebras

The Wikipedia page on Stiefel-Whitney classes includes the following paragraph: Over the Steenrod algebra, the Stiefel–Whitney classes of a smooth manifold (defined as the Stiefel–Whitney ...
4
votes
1answer
222 views

Chern Classes and Stiefel-Whitney Classes

I'm trying to understand the relationship between Chern classes and Stiefel-Whitney classes, and I came upon this problem (14-E) in Milnor-Stasheff's Characteristic Classes. We are asked to define ...
2
votes
0answers
73 views

Image of Thom Class under Sequence of Maps?

So I've been trying to do problems in Milnor & Stasheff's Characteristic Classes as a quick review, not having done anything with them in a while. However, I'm stuck on some parts in attempting ...
4
votes
0answers
136 views

Characteristic Class exercises

I tagged this as "homework" because my supervisor told me I need to be better at computing characteristic classes. The classic examples I can think of are tangent bundles and tautological line ...
3
votes
1answer
246 views

Determining the “positivity” or “negativity” of Chern class (number?) of zero-sets of homogeneous polynomials

If $\Omega$ is the curvature 2-form on a $n-$manifold, then I would think that the Chern classes (forms), $c_k$ are defined as, $det(I + \frac{it\Omega}{2\pi}) = \sum c_k t^k$ I would like to ...
9
votes
3answers
504 views

How to interpret the Euler class?

Although I (hardly) understand the formal definition of the Euler class, I have very little intuition of it. I understand that the Euler class of $E\to X$ is zero if and only if there is a section, ...
5
votes
1answer
171 views

Why are characteristic classes well-defined?

In the definition of characteristic classes for a complex vector bundle $E$ ober a topological space $X$, we consider some space $X_S$ and a continuous map $p: X \rightarrow X'$ such that $E$ is ...
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 ...
5
votes
2answers
199 views

Vanishing of the second Stiefel–Whitney classes of orientable surfaces

How does one see that the second Stiefel-Whitney class is zero for all orientable surfaces. For $S^2$ this can be seen by $TS^2$ being stably trivial, and for $S^1 \times S^1$ one can use $T (S^1 ...
2
votes
0answers
63 views

The characteristic class of a fibration is a fibre homtopy invariant

Let $p:E \to B$ be a fibration with fibre $F$ (denote the fibration $\xi$). Let us assume that the fibre is $(n-1)$ connected. There is a fundamental class $\iota_F \in H^n(F;\pi_n(F))$. We can define ...
2
votes
0answers
86 views

Question involving the Chern character from the book “Fibre Bundles”

On page 311 of Dale Husemöller's book Fibre Bundles in Theorem 11.6 he has the following commutative diagram $$\begin{array} & & K(BG)\\ &\nearrow &\downarrow\\ ...
4
votes
0answers
99 views

The action of the Steenrod algebra on $H^*(BU; \mathbb{Z}_p)$

By considering the classifying map $f \colon (\mathbb{C} P^{\infty})^n \rightarrow BU(n)$, its induced map on cohomology, and using the Cartan formula, we can derive the Wu formula for the action of ...
2
votes
1answer
96 views

Codimension 1 immersions and Stiefel-Whitney classes

From Milnor and Stasheff: If the $n$-dimensional manifold $M$ can be immersed in $\mathbb{R}^{n+1}$ show that each Stiefel-Whitney class $w_i(M)$ is equal to the $i$-fold cup product $w_1(M)^i$. ...
1
vote
1answer
122 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 ...
14
votes
2answers
424 views

How to understand the Todd class?

I am reading the article "K-Theory and Elliptic Operators"(http://arxiv.org/abs/math/0504555), which is about Atiyah-Singer index theorem. In page 14 the article discussed the Thom isomorphism: ...
0
votes
1answer
115 views

What is the second stiefel whitney class of SO(n)?

$\omega_2(SO(n))=?$, that is, What is the second stiefel whitney class of SO(n)?
5
votes
1answer
141 views

How can we detect the existence of almost-complex structures?

Any smooth $2n$-manifold $M$ comes with a well-defined map $f:M\rightarrow BGL_{2n}(\mathbb{R})$ (up to homotopy) classifying its tangent bundle. Since $GL_{2n}(\mathbb{R})$ deformation-retracts onto ...
3
votes
1answer
160 views

Does naturality imply isomorphism invariance for characteristic classes?

A property of characteristic classes $c$ is that $c(f^* E) = f^* c(E)$ where $E\to M$ is a bundle and $f^* E$ is the pullback of $E$ by some map $f: N\to M$. In Bott and Tu it is stated (for Chern ...
3
votes
1answer
103 views

Differing axioms for Stiefel-Whitney class

In Milnor and Stasheff, it is taken as part of the first axiom that all Stiefel-Whitney classes of a bundle vanish in dimensions greater than the rank of the bundle. However, in other sources this is ...
3
votes
1answer
191 views

What's the map $BU \times \mathbb{Z} \to \prod K(\mathbb{Z},n)$ representing the total Chern class?

Recall that complex topological $K$-theory is representable on reasonable spaces by the space $BU \times \mathbb{Z}$ (where $BU$ is a colimit of various infinite Grassmannians), and that the total ...
4
votes
1answer
253 views

Euler class and Vandermonde polynomial

I found the following in the wikipedia page for Euler class. «If the rank $r$ is even, then this cohomology class $e(E) \cup e(E)$ equals the top Pontryagin class $p_{r/2}(E)$. Under the splitting ...