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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Are these “Stiefel-Whitney numbers” invariants of the (smooth)topology of the total space?

Let $E$ be a smooth real vector bundle over a smooth compact $n$-dimensional manifold $M$. It is relatively easy to see that the Stiefel-Whitney classes $w_i(E)$ are not diffeomorphism invariants of ...
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Hirzebruch's $L$-polynomial and $\mathbb{C}P^n$

Hirzebruch's $L$-polynomial is the formal power series \begin{equation} L(x) = \frac{x}{\tanh x} = 1 + \frac{x^2}{3} + \cdots \end{equation} This defines a multiplicative sequence and a genus $L(M)$ ...
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Pontryagin class of a wedge product of vector bundles.

Let $E\to M$ be a real vector bundle over a differentiable manifold $M$ and let $p_{1}(E)$ denote its first Pontryagin class. I would like to know if there is any formula allowing to write ...
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Intuitive explanation of bordism invariant?

As the question suggests, what is an intuitive explanation of "bordism invariant"? I tried looking up some examples but they were drenched with jargon...
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Chern-Gauss-Bonnet theorem for even-dimensional manifolds with boundary

On the wikipedia page for the Chern-Gauss-Bonnet theorem it states that there is a generalization of the theorem for even-dimensional manifolds with boundary, but does not provide the relevant theorem ...
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Literature concearning characteristic classes?

It sounds the literature about characteristic classes is not very abundant (am I wrong?). Whenever I look for books dealing with this matter I'm always lead to the same material like the classical ...
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$w(\mathbb{R}P^q) = 1$ if and only if $q = 2^k - 1$ for some $k$

How do I show that $w(\mathbb{R}P^q) = 1$ if and only if $q = 2^k - 1$ for some $k$?
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Proving a $k$-multilinear symmetric map is invariant iff a condition is satisfied

In Huybrecht's book on complex geometry, he states the following lemma on page 193: Lemma 4.4.2: The $k$-multilinear symmetric map $P$ is invariant if and only if for all $B,B_1,\ldots, B_k \in ...
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Integral of generator $H^{2k}(\mathbb CP^k)$ over $\mathbb CP^k$ is 1

I'm trying to give a proof of the Hirzebruch signature theorem from a differentiable viewpoint (i.e. using de Rham cohomology). The proof works, except for one small step. We know that ...
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43 views

Vanishing of the first Chern class of a complex vector bundle

Suppose that $E\to M$ is a $\mathbb{C}^n$-bundle with a metric. This is equivalent to saying that there exists a chart $\{U_\alpha\}$ of $M$ and $\phi_{\alpha,\beta} \colon U_\alpha\cap U_\beta\to ...
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Orientability of $\gamma^n\oplus \gamma^n$ WITHOUT characteristic classes

I was curious to find an argument to show orientability of the $2n$-bundle $$\gamma^n\oplus \gamma^n$$ where $\gamma^n$ is the canonical $n$-bundle over the infinite grassmannians ...
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Prove that $w_{2n}(\gamma^n\oplus \gamma^n)\neq 0$

Let $\gamma^n$ be the canonical $n$-plane bundle over the infinite Grassmann manifold $G_n(\mathbb{R}^{\infty})$. I'm asked to prove that $$w_{2n}(\gamma^n\oplus \gamma^n)\neq 0$$ (exercise 9-A ...
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Introductions to Ehresmann connections and Chern-Simons forms

I am looking for introductory texts on Ehresmann connections and Chern-Simons forms. I seek detailed, hands-on presentation. Please, recommend sources that employ a differential forms approach rather ...
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45 views

underlying real vector bundle of a complex vector bundle

Let $\eta^\mathbb{C}$ be a complex line bundle. If the underlying $2$-dimensional vector bundle $\eta$ is not trivial as a real vector bundle, can we obtain that $\eta^\mathbb{C}$ is not trivial as a ...
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Formula for Stiefel-Whitney of tensor product

I need help for solving Ex. 7C from 'Characteristic Classes' by Milnor/Stasheff: The exercise asks to find a formula for the (total) Stiefel-Whitney class of $\xi^m\otimes\eta^n$ over a paracompact ...
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Relations on Stiefel-Whitney classes

Can arbitrary cohomology classes $w_1,\dots,w_n$ from $H^{*}(B,\mathbb{Z}_2)$ be Stiefel-Whitney classes of some bundle over the given base $B$ or there are some necessary relation which can be ...
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Does naturality for characteristic classes imply the classifying space is universal for them?

Let $G$ be a Lie group, $\mathfrak g$ its Lie algebra, $K$ its maximal compact subgroup. To every flat $G$-bundle $P$ over a smooth manifold $M$ I can associate a homomorphism $w_P: H^*(\mathfrak g, ...
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The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?

In Milnor/Stasheff Characteristic classes on page 180 there is a statement (inside a proof) that I don't fully understand. Let $\tilde{G}_k$ be the oriented real grassmanian, $e$ its Euler class: ...
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58 views

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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Constructing commutative diagram up to homotopy from a smooth map.

This is from the proof of Theorem 20.7 in "Characteristic classes" by Milnor and Stasheff. Let $f : M^n \rightarrow S^r$ be a smooth map where $M$ is smooth $n$-dimensional manifold and $S^r$ is ...
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Stiefel Whitney numbers of a $3$- manifold are $0$

In Milnor's book on Characteristic Class he asks to prove that (Problem 11-D) all Stiefel Whitney numbers of a $3$- manifold are $0$. I can show $w_{3}=0$ as $\chi (M)=0$. From dimension consideration ...
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Pontryagin classes of a tensor product of bundles

This question is related to " How to calculate characteristic classes of tensor products? " but interested in the Pontryagin classes instead of $c_1$. Specifically, given two real vector bundles $E$, ...
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Cobordism of two manifolds [closed]

Is $\mathbb RP^4 \times \mathbb RP^{12} \times \mathbb RP^{15}$ cobordant to $\mathbb RP^6 \times \mathbb RP^{9} \times \mathbb RP^{9} \times \mathbb RP^{7}$?
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Chern classes mod 2 equal Stiefel-Whitney classes via Milnor/Stasheff language

I'm having truble with Exercise 14B of Milnor/Stesheff Characteristic classes: prove that the total Chern class of a comple bundle is mapped to the Stiefel-Whitney class by the coefficient ...
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Stiefel-Whitney Numbers of $\mathbb{R}P^2\times \mathbb{R}P^2$

I'd like to calculate the Stiefel-Whitney numbers of $\mathbb{R}P^2\times\mathbb{R}P^2,$ but don't know how to. My first instinct was to say that the tangent bundle is isomorphic to the product of ...
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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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100 views

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 ...