Characteristic classes are invariants of bundles living in the cohomology of the base. The most common examples of characteristic classes are the Chern, Stiefel–Whitney, and Pontryagin classes.

learn more… | top users | synonyms

14
votes
2answers
556 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: ...
13
votes
2answers
365 views

Where do Chern classes live? $c_1(L)\in \textrm{?}$

If $X$ is a complex manifold, one can define the first Chern class of $L\in \textrm{Pic}\,X$ to be its image in $H^2(X,\textbf Z)$, by using the exponential sequence. So one can write something like ...
12
votes
3answers
908 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, ...
12
votes
2answers
243 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 $\alpha\in H^2(M,\mathbb{Z})$: ...
8
votes
1answer
266 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 ...
6
votes
1answer
187 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 ...
6
votes
1answer
131 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 ...
6
votes
1answer
226 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} ...
6
votes
1answer
168 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
1answer
172 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 ...
6
votes
1answer
114 views

Questions on Chern characters.

Let $X$ be a complex manifold and $\mathcal{O}_p$ a skyscraper sheaf. How can one compute the Chern character $ch(\mathcal{O}_p)$? For vecoto bundles, we have $ch(V)\cup ch(W)=ch(V\otimes W)$, but ...
6
votes
0answers
39 views

Chern classes of free quotient manoflds

Let $X$ be a compact complex manifold. Assume that a finite group acts on $X$ freely. Then the quotient $X/G$ is again a compact complex manifold. I wonder if there is a good way to compute Chern ...
6
votes
0answers
405 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 ...
5
votes
2answers
282 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 ...
5
votes
1answer
188 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 ...
5
votes
2answers
424 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 ...
5
votes
1answer
125 views

Computing characteristic numbers of homogeneous spaces

I apologize in advance if this is a bad question. I would like to prove or refute a conjecture about the vanishing of characteristic numbers of homogeneous spaces, and to this end am looking (and ...
5
votes
1answer
134 views

First Pontryagin class on real Grassmannian manifold?

I wonder if real Grassmannian manifold $SO(p+q)/SO(p) \times SO(q)$ have nontrivial first Pontryagin class? I only have physics background and know really little about characteristic class theory.
5
votes
0answers
111 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 ...
4
votes
1answer
281 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 ...
4
votes
1answer
311 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 ...
4
votes
1answer
50 views

Working out an example of a Chern class

I'm trying to understand page 161 of Fulton's "Young tableaux" in an explicit example. I'm looking at flags in $\mathbb{C}^4$, which I think of as flags in $\mathbb{CP}^3$ (and I'm really just able ...
4
votes
1answer
67 views

Representation of an oriented manifold as a set of common zeroes of smooth functions

Let $M$ be an arbitrary oriented smooth manifold of dimension $m$. Is it always diffeomorphic to a sumbanifold in ${\mathbb R}^n$ (with some $n$) defined as a set $X$ of common zeroes of $n-m$ smooth ...
4
votes
1answer
107 views

On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.

This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where ...
4
votes
2answers
88 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 ...
4
votes
1answer
245 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 ...
4
votes
1answer
160 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 ...
4
votes
0answers
30 views

2-forms represented by a first Chern class?

Let $M$ be a complex manifold and $\omega$ be a 2-form on $M$. Is there a good way to see whether $\omega$ is represented by the first Chern class of a line bundle on $M$? In other words, when is it ...
4
votes
0answers
197 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 ...
4
votes
0answers
174 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 ...
4
votes
0answers
118 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 ...
4
votes
0answers
257 views

Intuition for multiplicative sequences

I have recently been reading about multiplicative sequences and genera from a couple of different sources, most notably "Spin Geometry" by Lawson and Michelsohn and "Characteristic Classes" by Milnor ...
3
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 ...
3
votes
1answer
143 views

Euler characteristic is equal to self-intersection number of zero-section?

As I recall (from Guillemin and Pollack "Differential Topology") the Euler characteristic of a (for my purposes, compact and oriented) smooth manifold X is defined as $\chi(X)=I(\Delta,\Delta)$, ...
3
votes
1answer
180 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
173 views

Intuition of Chern-Weil theory

Let $ P \rightarrow M$ be a $G$-principal bundle. The lie algebra of $G$ is $\frak{g}$ and $P$ has connection form $\omega \in H^1(P,\frak{g})$ and curvature form $\Omega \in H^2(P,\frak{g})$. We ...
3
votes
1answer
105 views

Translation of french paper into English

I am currently reading a mathematical paper in french and I am not sure how to translate the following sentence: "On suppose que la premiere classe de Chern $c_1(N)$ est $p\alpha$ ou $p$ est un ...
3
votes
1answer
209 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 ...
3
votes
2answers
233 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 ...
3
votes
1answer
79 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 ...
3
votes
1answer
338 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 ...
3
votes
1answer
111 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
139 views

Thom class: Why are the two definitions equivalent?

We know that the Thom class $\tau_W$ is defined on a disk bundle $W\rightarrow L$, where $L$ is a $p$-dimensional manifold and the rank of $W$ is $k$. Let $[W]_0$ denote the fundamental class of the ...
3
votes
1answer
143 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
1answer
144 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 ...
3
votes
0answers
55 views

What is a $c_1$-map for Riemann-Roch theorems?

Atiyah and Hirzebruch define a $c_1$-map to spell out Riemann-Roch theorem for (compact and connected) smooth manifolds. The definition is following: a map $f:Y \to X$ is called a $c_1$-map if we are ...
3
votes
0answers
54 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 ...
3
votes
0answers
156 views

Two Grassman manifold problems from Milnor and Stasheff's book

I'm stuck on the following two problems in Milnor and Stasheff's book Characteristic Classes. Really can't get my head around this material and I'm hoping that more worked examples would help. Even ...
2
votes
1answer
125 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 $w_1(M)^i$. ...
2
votes
1answer
120 views

The Stiefel-Whitney classes of Cartesian product

I am reading the book of characteristic classes by Milnor-Stasheff, and I have a problem with the exercise 4-A: Show that the Stiefel-Whitney classes of a Cartesian product are given by ...