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.

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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, ...
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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: ...
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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 ...
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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 $\alpha\in H^2(M,\mathbb{Z})$: ...
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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 ...
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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 ...
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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 ...
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Relation between Stiefel-Whitney class and Chern class

A complex vector bundle of rank n can be viewed as a real vector bundle of rank 2n. From nLab, we have that the second Stiefel-Whitney class of the real vector bundle is given by the first Chern class ...
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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 ...
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Why $\mathbb{RP}^2$ can not be embedded to $\mathbb{R}^3$?

Is there any answer of this question around basic theory of differentiable manifolds?
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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 ...
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Chern-Weil: why do we divide by $2\pi$?

So here's a somewhat incoherent question. To define characteristic classes in the Chern–Weil way, one takes a curvature form $\Omega$ on a vector bundle $E \to M$ and an invariant polynomial ...
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What can we say about a vector bundle with trivial unit sphere bundle?

If you want to avoid the back story to this question, feel free to skip the paragraphs between the horizontal lines. Yesterday Georges Elencwajg asked me the following question (I'm paraphrasing): ...
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Relations of Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, Wu class. [closed]

We have a quite some characteristic classes: Chern class, Stiefel-Whitney class, Pontryagin class, Euler class, Wu class, etc. I wonder whether some math experts can use simple words and basic ...
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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 ...
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Topological invariance of chern classes

Are Chern classes topological invariants? To be more precise: Given two complex manifolds $M$ and $N$. Does a homeomophism $f:M\to N$ map Chern classes to Chern classes?
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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 $\chi(X)=I(\Delta,\Delta)$, ...
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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 ...
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Why is the constant relating the chern class and curvature form always $2\pi i$?

I'm reading Milnor's book on Characteristic Classes. In Appendix C, Milnor shows the invariant polynomial of the curvature form and the Chern class differ by powers of $2\pi i$. He first shows that ...
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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.
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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} ...
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Intersection theory proof of the poincare hopf theorem.

Suppose that $M$ is a connected compact oriented smooth manifold, and $X:M\rightarrow TM$ a vector field with isolated zeros. Then if $Z$ is the zero set of $X$ (a $0$-dimensional oriented manifold), ...
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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 ...
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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 ...
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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 ...
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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 ...
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Has anyone seen this combinatorial identity involving the Bernoulli and Stirling numbers?

Does anyone know a nice (combinatorial?) proof and/or reference for the following identity? $$\left( \frac{\alpha}{1 - e^{-\alpha}} \right)^{n+1} \equiv \sum_{j=0}^n \frac{(n-j)!}{n!} |s(n+1, ...
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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 ...
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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 ...
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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 ...
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Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the ...
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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 ...
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Elementary proof of the fact that any orientable 3-manifold is parallelizable

A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors ...
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If the top Stiefel-Whitney class of a compact manifold is nonzer0, must there be another non-vanishing Stiefel-Whitney class?

I was trying to collect some examples of Stiefel-Whitney class computations, just to make myself more familiar with them. It seems from my (relatively short list) that if the top Stiefel-Whitney ...
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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 ...
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Chern classes via connections

Let $M$ be a smooth real manifold and $B$ an Hermitian vector bundle over it. Then one can define Chern classes as $$c(B)=\sum c_i(B)t^i=\det \left( I+\frac{it\Omega}{2\pi} \right) \in ...
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Manifolds with vanishing Stiefel-Whitney classes but are not stably parallelizable

It is known that if a manifold is stably parallelizable, then it's Stiefel-Whitney classes must vanish. Is the converse true? Note that we know that the converse cannot hold if stably parallelizable ...
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Integrality of the $L$-genus for a smooth manifold

For a compact, oriented, smooth manifold $M^{4k}$, the Hirzebruch signature theorem gives the signature $\sigma(M)$ in terms of a polynomial $P_k$ in the Pontryagin numbers of $M$ whose coefficients ...
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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 ...
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Determining an explicit line bundle over surface

The following is a explicitly defined complex line bundle $E\to\Sigma$ over a closed surface: View $\Sigma$ as a subset of $\mathbb{R}^3$ and consider its Gauss map $n:\Sigma\to S^2$ given by ...
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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 ...
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Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
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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 ...
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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 ...
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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 ...
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Milnor's definition of bundle map in “Characteristic Classes”

In chapter 3 of "Characteristic Classes", Milnor defines bundle maps, requiring them to map fibers isomorphically onto fibers. Why not merely require homorphisms on fibers? (e.g., for the given ...
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Second Chern class of $TS^2$

The induced metric on a sphere may be given by, $$ds^2 = d\theta^2 + \sin^2\theta\, d\phi^2$$ By using Cartan's method of moving frames, one can compute the curvature 2-form in an orthonormal basis ...
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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 ...
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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 ...
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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 ...