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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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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Chern class of line bundle and vector bundle

Let $L$ is a Line bundle and $E$ a vector bundle of rank $r$ then how can we prove that $$c_1(L\otimes E)=rc_1(L)+c_1(E)$$ where here $c_1$ means first chern class
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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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Basic question on almost complex structures and Chern classes of homogeneous spaces

Toward the end of "Characteristic Classes and Homogeneous Spaces, III," Borel and Hirzebruch prove that given a compact Lie group $G$ and toral subgroup $T$ (no restriction on rank), one has $w(G/T) = ...
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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 ...
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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 ...
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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 ...
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What are other examples of characteristic numbers?

Be warned, this may be a ridiculous question. I understand characteristic classes of principal $G$-bundles (and associated vector bundles) over a space $X$ arise from the classifying maps $f\colon X ...
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Non-existence of nowhere vanishing vector field on even dimensional sphere

I want to show that for even dimensional spheres, there does not exist a nowhere vanishing vector field, namely a non-trivial cross section of its tangent bundle. I am wondering how elementary the ...
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Exercises - “From calculus to cohomology”

I am reading Madsen's book From calculus to cohomology and I've found it doesn't have any (explicit) exercises at the end of each section. I'd like to know a few books where I can find some problems ...
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Nonaxiomatizability one-dimensional vector spaces [duplicate]

Prove, class of all one-dimensional vector spaces over R isn't axiomatizable in the signature σ=< +, α, 0>, α* is single function vector multiplication by a scalar α of R.
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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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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 ...
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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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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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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 ...
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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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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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Is there any standard N-sphere that has non-trivial first Pontryagin class?

I am wondering if there is a standard $N$-sphere that has non-trivial first Pontryagin class on its tangent bundle $TS^n$and frame bundle $FS^n$? I know that only $S^4$ has non-trivial $H^4(S^n, R)$ ...
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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 ...
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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 ...
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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 ...
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Chern character of a sheaf with support of given dimension

Let $\mathscr{F}^\bullet$ be a complex of coherent sheaves on a smooth projective variety $X$. Suppose that the support of $\mathscr{F}^\bullet$ (the union of the supports of the cohomology sheaves, ...
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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 ...
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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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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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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Chern Character Isomorphism for non-CW complexes

Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \mathrm{H}^\ast(X; \mathbb{Q})$, where $\mathrm{H}^\ast$ denotes ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $T\mathbb{C}P^n$: $$ ...
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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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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 ...
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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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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 ...
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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 ...
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A question on Chern character computation

Let $C$ be a smooth curve in a complex threefold $X$. How can I see that $$ \mathrm{ch}(\mathcal{O}_C)=(0,0,[C],\chi(\mathcal{O}_C))\in H^0\oplus H^2 \oplus H^4\oplus H^6, $$ where $H^0\cong ...
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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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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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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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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 ...