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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Euler Classes, Chern Classes, $S^2$ Bundles, and $CP^1$ Bundles

I am just starting out learning about characteristic classes (Euler, Chern, etc.) from Bott and Tu's book, and I had the following question. Let $E$ be an oriented $S^2$ bundle over $M$ with ...
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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 ...
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Relative cohomology of a vector space module non-zero vectors

I am trying to explicitly compute the relative cohomology groups $H^m(\mathbb R^n,\mathbb R^n_0;\mathbb Z)$, where $\mathbb R^n_0$ is all the non-zero vectors in $\mathbb R^n$. I think that the answer ...
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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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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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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 ...
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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 ...
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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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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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Euler class is odd under orientation, thus its integral over a manifold will be even.

I learned a statement from others: "Euler class is odd under orientation, thus its integral over a manifold $M$ is even." I cannot fully appreciate it, can someone show this explicitly? The ...
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Calculate the Wu class from the Stiefel-Whitney class

The total Stiefel-Whitney class $w=1+w_1+w_2+\cdots$ is related to the total Wu class $u=1+u_1+u_2+\cdots$: The total Stiefel-Whitney class $w$ is the Steenrod square of the Wu class $u$: ...
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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 ...
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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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the Pontryagin number for 4-dim surface bundle

Corollary 1.8 in arxiv.org/pdf/1103.0218 implies that the Pontryagin number for a 4-dim surface bundle is non-zero only when the surface has a genus $g>2$. I would like to ask what is the minimal ...
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Geometric picture of Stiefel-Whitney class of a tangent bundle?

The first Stiefel-Whitney class $w_1$ of a tangent bundle of a manifold $M$ has a simple geometric picture: if there is a loop that the orientation of the tangent space reverses, then $w_1\neq 0$ and ...
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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-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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Naive question: what good are characteristic classes of principal bundles?

I recently read a development of characteristic classes on principal bundles through curvature forms and the Chern–Weil homomorphism. Unfortunately, this exposition concluded without listing any ...
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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 ...