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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Tangent bundle of manifold with no odd dimensional sub-bundles

First, a preliminary remark: The Whitney sum of two vector bundles is orientable. I saw this statement somewhere and was wondering if it's true. In particular, it's easy to show that ...
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$\mathbb{R}P^4$ and $\mathbb{R}P^6$ do not admit fields of tangent $2$-planes

See the related question here. This is the second part of question 4-C in Milnor and Stasheff's book on characteristic classes. In the solution to the first part, we rely on the fact that having a ...
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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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Examples with zero first Stiefel-Whitney class and nonzero second Stiefel-Whitney class

What's the simplest/most concrete vector bundle you can think of that has zero first Stiefel-Whitney class but non-zero second? That would be the simplest space that doesn't have spinors. (See Spin ...
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The second Chern class of line bundle

This is from the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle Ch2 Lemma 1. Let L be a line bundle on the effective divisor D$\to$X and j:D$\to$X be the inclusion. Then ...
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Chern class of complex vector bundles

Let $\xi$ be an $n$-dimensional complex vector bundle. It is claimed that the Chern class of $\xi$ is $$ c(\xi)=(1+x_1)\cdots (1+x_n), $$ $|x_k|=2$, $c_j(\xi)$ is the $j$-th symmetric polynomial of ...
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Stiefel-Whitney classes with Z-coefficients

Stiefel-Whitney classes are defined (for example, in Milnor's Characteristic classes) as elements of the cohomology groups with $\mathbb{Z}_2$-coefficients as follows. $$w_i(E)=\phi^{-1} Sq^i(u)$$ ...
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Compute a chern class $c(K^n)$ for a non-singular algebraic hypersurface $K^n$ of degree $d$ in $P^{n+1}(C)$.

This is Problem 16-D in Characteristic classes by John W. Milnor and James D. Stasheff. Problem 16-D) If the complex manifold $K^n$ is complex analytically embedded in $K^{n+1}$ with dual ...
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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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Quick question: Tensoring a 2-torsion line bundle with a rank 2 nonsplit extension over a curve of genus 1

Everything is complex algebraic. Over a curve $C$ of genus $1$, let $V$ be a rank 2 vector bundle with $\deg \det(V)=1$, which is a nonsplit extension of $\mathcal{O}_C$ and $\mathcal{O}_C(p)$, where ...
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Stiefel-Whitney class of complex projective spaces

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$? Let $a_m$ be the maximal integer such that the $a_m$-th dual ...
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Constructing complex line bundles on orientable smooth manifolds

This questions ask how to construct a complex line bundle over a smooth compact orientable manifold without boundary starting with an n-2 dimensional orientable sub manifold without boundary. The ...
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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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What is the Chern class of the Kernel of a projection map after taking a blowup?

Let $S:= \mathbb{CP}^1 \times \mathbb{CP}^1 $ and $\tilde{S}$ the blowup of $S$ at one point. Let $a_1, a_2$ be generators for the cohomology $H^*(S, \mathbb{Z})$ and let $a_1, a_2$ and $E$ be the ...
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Identity used to prove the Chern-Weil theorem

I'm reading the proof of Chern-Weil theorem found in Nakahara's second edition book (page 422) but I got stucked at the very beginning of the proof. It says that from the identity ...
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What is the poinacre dual of the projectivization of a line bundle inside the projectivization of the sum of two line bundles?

Let $S:= \mathbb{CP}^1 \times \mathbb{CP}^1$. Let $TS \approx L_1 \oplus L_2$, where $L_1$ and $L_2$ are the pullback of $T\mathbb{CP}^1$ wrt to the respective projection maps. Note that ...
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given a specific vector bundle how to see whether the first Pontryagin class is zero

Let $Z_2$ be the group with $2$ elements. Let $a\in Z_2$ be the nontrivial element. Let $S^n$ be the $n$-sphere. Let $C(S^n,2)=\{(x,y)\in S^n\times S^n\mid x\neq y\}$. Let $a$ act on $C(S^n,2)$ by ...
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a question on oriented bundles and Euler class

In Characteristic classes, J. Milnor, J. Stasheff, Prop. 9.7, it is proved that: if the oriented vector bundle $\xi$ possesses a nowhere zero cross section, then the Euler class $e(\xi)=0$. I want ...
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A complex line bundle is trivial if and only if the first Chern class is zero

Let $\xi$ be a complex line bundle over a CW-complex $B$. I want to prove that $\xi$ is trivial if and only if $c_1(\xi)=0$. My attempt: Suppose $c_1(\xi)=0$. Then the Euler class $e(\xi)=0$. Since ...
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ways to see whether the Pontryagin class of a quaternionic line bundle over a CW-complex is zero

the first pontryagin class of a quaternionic line bundle over a CW-complex is zero if and only if the quaternionic line bundle is trivial or not? Let $\xi^\mathbb{H}$ be a given quaternionic line ...
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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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chern class of complex line bundle

Let $\xi:=(\mathbb{C},E,p,B)$ be a complex line bundle, where $B$ is a manifold or CW-complex. How to determine whether the first Chern class $c_1(\xi)=0$ or non-vanishing?
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integral cohomology of real grassmannians

I have obtained that the cohomology rings $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]. $$ Also $$ H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
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The tangent bundle of $\mathbb{CP}^1$ is not isomorphic to its dual.

This question is related to this one but is not a duplicate. In " Characteristic Classes" by Milnor and Stasheff on pages 167-168, the authors give a brief argument about why the tangent bundle of the ...
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the top chern class of the holomorphic tangent bundle is the euler class

Is the following true? Let X be a complex manifold of complex dimension d and let V denote its holomorphic tangent bundle (ie it's $T^{1,0} \subset T \otimes_R C$, where T is the tangent bundle of ...
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Reference request: Zero set of global section

Things are in the complex algebraic setting. Assume that a vector bundle $V$ of rank $n$ over a $\mathbb{P}^n$ has a global section $\sigma$. Is it true that the zero set of $\sigma$ is a ...
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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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cohomology of classifying space of cyclic group

(1). Let $p$ be a prime number. Let $B\mathbb{Z}_p$ be the classifying space of the discrete group $\mathbb{Z}_p$. How to obtain $$ H^*(B\mathbb{Z}_p;\mathbb{Z}_p)=\mathbb{Z}_p[t]\otimes \Lambda[e]? ...
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K-theory formulation of the index theorem

The Atiyah-Singer index theorem is one of the most important results in twentieth century's mathematics. It states that for an elliptic differential operator $D$ on a smooth, oriented compact manifold ...
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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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cohomology of complex grassmannians and quaternion grassmannians (the finite dimensional case)

Let $G_k(\mathbb{C}^N)$ be the complex grassmannian manifold. Let $G_k(\mathbb{H}^N)$ be the quaternion grassmannian manifold. Let $p$ be a prime integer (we may also impose extra conditions, such as ...
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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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cohomology homomorphism between grassmannians induced by inclusion

Let $i:G_k(\mathbb{R}^n)\to G_k(\mathbb{R}^\infty)$ be inclusion of grassmannians. Then $H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]$. $ ...
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cohomology homomorphism induced by classifying map

Let $\xi=(E,p,B)$ be an $n$-dimensional vector bundle. Let $f: B\to G_n(\mathbb{R}^\infty)$ be the classifying map. Let $f^*: ...
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Reference request: second Chern class of P^2

I have heard that $c_2(T_{\mathbb{P}^2})=e(\mathbb{P}^2)$. What's the general result and where can I read about it? Thanks.
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cohomology of finite dimensional grassmannians

What is the cohomology algebra of finite dimensional grassmannian $$ H^*(G_k(\mathbb{R}^N);\mathbb{Z}_2)? $$ $$ H^*(G_k(\mathbb{C}^N);\mathbb{Z}_p)? $$ $$ H^*(G_k(\mathbb{H}^N);\mathbb{Z}_p)? $$ I ...
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The first Chern class of projective line $\mathbb{CP}^1$

I am studying the Chern class using by some textbooks and lecture notes. One day, I found an example of the first Chern class of $\mathbb{CP}^1$. Let $\xi$ be a tautological line bundle of ...
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The smallest $n> 0$ with the nonzero $n$th Stiefel-Whitney class is a power of 2 when total Stiefel-Whitney class is not trivial.

This is the Problem 8-B form the characteristic classes by John W. Milnor and James D. Stasheff. [Problem 8-B]. If the total Stiefel-Whitney class $w(\xi) \neq 1$, show that the smallest $n>0$ ...
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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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Characteristic class integral: on what manifold does $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$ hold?

Characteristic class integral: when does the equality hold $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$, on what manifolds? Here $c_1$ is the first Chern class. Here $w_2$ is the 2nd ...
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Weyl Groups/Borel

Could someone tell me where to find a proof of the following statement that I found in some notes about characteristic classes I was reading? If $G$ is a compact connected Lie group with maximal ...
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Stably equivalent bundles and stable homotopy classes of maps

We know that there is a one-to-one correspondence between isomorphism classes of principal $G$-bundles over a base space $M$ and homotopy classes of maps $M \to BG$, where $BG$ is the classifying ...
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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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A question about characteristic classes

I have a map $\phi:BO(1)^n\rightarrow BO(n)$ which is given by sending any $n$-tuple in $BO(1)^n$ to an $n$-plane through the origin. Thus, this induces a group action on the symmetric group $S_n$ ...
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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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Vector bundle base space map

Is it true, and if it is, is there some easy way to see the following? Suppose that $\xi = (\pi, E, B)$ is an $n$-vector bundle with $B$ paracompact but not necessarily compact. Is there a base ...
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classifying map of associated bundles

Let $G\leq GL(\mathbb{R}^n)$ be a group and $\xi$ be a principal $G$-bundle over a space $X$. Let $\eta=\xi[\mathbb{R}^n]$ be the associated vector bundle of $\xi$. Let $f_\xi: X\to BG$ be the ...
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If we construct an euler class, why is it that it behaves functoriality (Naturality)? That is:

Here is a very simple question that I can't seem to figure out for some reason. If we construct an euler class, why is it that it behaves functoriality (Naturality)? That is: $e(F) = f*e(E)$
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cup product of stiefel-whitney class

Let $\xi$ be a vector bundle. Let $w(\xi)$ be the total Stiefel-whitney class. Let $\bar w$ be the dual Stiefel-whitney class. In John Milnor's Characteristic class book, page 40-41 Chap.4, ...
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Analogue in algebra for characteristic classes?

By Swan's Theorem, we know that projective modules over a ring are an algebraic analogue of vector bundles over a base space. Is there some sort of cohomology theory of rings (or modules? or schemes, ...