Questions tagged [characteristic-classes]

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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$H^i\left(BO_m ; \mathbb{Z}_2\right) \cong H^i\left(B O_n ; \mathbb{Z}_2\right)$ for $i \leq m$ and $i \leq n.$

The notes I am reading say that groups in the title are isomorphic. Could someone explain to me why it is the case? Here by $BO_n$ I mean the infinite Grassmann manifold of $n$-dimensional subspaces. ...
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Cohomology ring of $BPin(n)$ with Z coefficients

I can't find references for the characteristic classes of $Pin(n)$ with coefficients in $\mathbb Z$. What is the cohomology ring $H^*(BPin(n); \mathbb Z)$ ? Is it true that $H^*(BPin(2); \mathbb Z)\...
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Integral Stiefel-Whitney classes and stable almost complex structure

Question: Let $\xi$ be a real vector bundle over a space $B$. Suppose $\xi$ admits stable almost complex structure, i.e., $\xi\oplus \epsilon^k$ admits almost complex structure for some $k\geq 0$, ...
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Euler class of a principal $SO(2)$-bundle over a lens space

Note that for a manifold $X$, isomorphism classes of principal $SO(2)$-bundles over $X$ are classified by their Euler classes in $H^2(Z;\Bbb Z)$. Now consider a lens space $L(p,q)$; we have $H^2(L(p,q)...
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How to compute Characteristic Classes from scratch and get which structure it capture

For our Differential Geometry II course we are strictly follow Differenetial Geometry, Loring W. Tu. Currently, I am reading chapter $5$, Vector bundles and Characteristic Classes. I feel the author ...
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If Chern classes are only defined for vector bundles, why can a $U(1)$ principal bundle have associated Chern classes?

When people talk about Chern classes of $U(1)$ principal bundles, do they really mean the Chern class of the associated vector bundle? As far as I can understand, Chern classes exist only for complex ...
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Euler class of a tensor product bundle

I want to prove the result: $$c_1(L\otimes L')=c_1(L)+c_1(L')$$ where $L,L'$ are complex line bundles and $c_1(L):=e(L_{\mathbb R})$. My definition of Euler class is given by the pullback by zero ...
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Chern-Weil theory in terms of pullback along classifying map

Let $G$ be a semisimple real Lie group and $M$ be a smooth manifold. The map on cohomology $H^{\ast}(BG,\mathbb{C}) \rightarrow H^{\ast}(BT,\mathbb{C})\simeq \mathbb{C}[\mathfrak{h}]$ induced by the ...
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Proving $c_1(L \otimes L') = c_1(L) + c_1(L')$ for line bundles.

I'm trying to prove this seemingly innocent result that for line bundles $L $and $L'$ the Chern class of the tensor product bundle is given by $$c_1(L \otimes L') = c_1(L) + c_1(L').$$ I managed to ...
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Pullback bundle and Chern classes

The following is from Bott and Tu. If $f : M \to N$ is a map between two manifolds and $E$ is a complex bundle over $N$, then the pullback $f^{-1}E$ is a bundle over $M$. If the Chern classes of $E$ ...
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First Chern class identically zero?

Let $E \to M$ be a complex vector bundle over a manifold $M$ with a connection $\nabla$. Denote by $\Omega$ the curvature matrix of $\nabla$. The $k$'th Chern class is then given by $$ c_k(E)= [f_k(i/...
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Existence of non-vanishing section of normal bundle (codimension $\geq 2$)

Suppose $M^n \hookrightarrow P^{n + k}$ is an isometric embedding of a manifold $M$ in an arbitrary ambient manifold $P^{n+k}$ for $k \geq 2$. Does there exist a global, non-vanishing section of the ...
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Equivalent definitions of Chern classes

I have met two different definitions for Chern classes and I am wondering how are they all the same? The first one is from Jöran Schlömer's book and they define the Chern class of a complex vector ...
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Defining Pontrjagin numbers

I'm reading about Pontrjagin numbers and I have difficulties understanding how they are defined. I have the following definition. Let $E$ be a vector bundle of rank $r$ over a compact manifold $M$ of ...
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How to understand all characteristic classes as generators of classifying space cohomologies

Context: I'm somewhere in the middle of my study of differential geometry and starting to learn about characteristic classes. I like to have a general intuitive understanding of a concept before ...
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multiple for first fractional Pontryagin class

Consider the Whitehead tower of the classifying space of the special orthogonal group $BSO(n)$. $$\cdots\to BString(n)\to BSpin(n)\to BSO(n)$$ There is an obstruction to lifting classifying maps into $...
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Relation between the first Pontryagin class $p_1$ and the first Stiefel Whitney class $w_1$ as a tangential structure of manifold M

In Characteristic Class, let us define tangential structure of manifold M such as tangent bundle TM. Is there a difference between the Stiefel Whitney class $w_1 =0$ and the first Stiefel Whitney ...
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Simplicial representative of the Thom class

Assume $p \colon Q \to P$ is a piecewise linear $n$-dimensional disc bundle. Let's consider triangulations $L$ and $K$ of $Q$ and $P$ in which $p$ is simplicial, and the subpolyhedron of spheres in $Q$...
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Finding the Chern character of a curve on the quintic threefold

Let $X$ be the quintic threefold in $\mathbb{CP}^{4}$ defined by the vanishing of the Fermat polynomial $$ x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}+x_{5}^{5} $$ and consider a rational curve $D$ on $X$....
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Clarification on the term

I came across the following statement when reading Section 4 of this paper: For an oriented 2-plane field $\xi$, let $d(\xi) \in \mathbb{Z}$ denote the divisibility of the Chern class, so that $c_1(\...
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Chern and Pontryagin classes of manifold constructed as compactification of vector bundle

I'm trying to compute the total Chern class of the tangent bundle of a manifold constructed as a "compactification" or "projective completion", but I'm not sure I quite have all ...
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About assertion in Thom isomorphism theorem

On page 114 of Milnor's book https://aareyanmanzoor.github.io/assets/books/characteristic-classes.pdf We have the next theorem He says that $H^i(E,E_0,\mathbb{Z}_2)=0$ for $i<n$ as a consequence ...
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Does a compact Kahler manifolds with negative first Chern class admits any nontrivial holomorphic vector field?

Let $M$ be a compact Kahler manifold with $c_{1}(M)\lt 0$. It seems that I can prove the following claim: there is no nontrivial holomorphic vector field on $M$. Here is my proof: let $T^{1,0}M$ be ...
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Integral first Chern class of the line bundle associated with a character

Let $X$ be a connected complex projective manifold, $\chi:\pi_1(X)\to S^1$ be a character of the fundamental group of $X$. Then $\chi$ induces a local system $\mathcal{L}_{\chi}$ of rank $1$ on $X$ ...
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Is a vector bundle which is trivial on a smooth submanifold trivial in a neighbourhood?

Let $E$ be a (real or complex) vector bundle on a smooth (possibly compact) manifold $M$. Suppose $N \subset M$ is a closed submanifold, such that $E|_N$ is trivial. Does there exist an open ...
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Oriented sphere bundle with section implies Euler class vanishes (using global angular form)

This exercise is 11.13 from Bott–Tu: Use the existence of the global angular form $\psi$ to prove that if an oriented sphere bundle $E$ has a section, then its Euler class vanishes. Now $d\psi=-\pi^*e$...
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Computing the total Pontrjagin class of quaternionic projective space

This is Problem 20-A in Milnor-Stasheff's book Characteristic Classes: Let $\tau$ be the tangent bundle of $\mathbb{P}^m(H)$, where $H$ is the quaternion algebra. $\gamma$ is the tautological bundle ...
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The existence of fields of tangent $2$-planes on real projective spaces

I'm interested in the following problem: When does $\Bbb RP^n$ admit a field of tangent $2$-planes? (Please assume $n>2$.) A field of tangent $k$-planes is a subbundle of $T\Bbb RP^n$ of rank $k$....
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Question about the Euler class

Let $M$ be a compact orientable $n$ dimensional smooth manifold without boundary, and $e\in H^n_{dR}(M)$ denote the Euler class of the tangent bundle $TM$. We have that \begin{align*} \int_Me=\chi(M) \...
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Doubt about splitting principle

I have a doubt about splitting principle. We know that if $E\to M$ is a complex vector bundle of rank $k$, then there exist a manifold $T=T(E)$ and a proper smooth map $f:T\to M$ such that $f^*:H^*(M)...
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The proof of Lemma11.9 and theorem 11.14 (Wu) (Milnor & Stasheff)

Lemma 11.9(page 127)showed the relation between diagonal class $u''\in H^n(M\times M)$ and fundamental class $\mu\in H_n(M)$ for a compact manifold $M$ with dimention n: $$u''/\mu=1$$ where $u'':=u'|_{...
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The proof of Duality theorem ( Milnor & Stasheff )

At the beginning of the proof(theorem 11.10 page 128), it says that it follows easily that the diagonal class can be expressed as r-fold sum \begin{align*} u''=b_i\times c_i+\ldots+b_r\times c_r \...
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Is there a geometric picture between Chern class and Chern character?

I'm new to the characteristics classes. When I learn the definition of Chern class and Chern character, the total Chern class is defined as: $$c(\mathcal F)=\det(I+\frac{i \mathcal F}{2\pi}),$$ where $...
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Stiefel-Whitney classes in Čech cohomology

Let M be a smooth manifold and Let G be a Lie group. We have the sheaf $\mathcal{F}_G=(F,\rho)$ of groups defined by $$F(U)=C^{\infty}(U,G)$$ for open sets $U\subset M$ (in particular set $F(\emptyset)...
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complexification of real vector bundles on $S^4$ and $S^8$

Prove there is a surjection $\tilde{KO}(S^8) \to \tilde{K}(S^8)$.(similarly for $S^4$) My idea: It is equivalent to show in every class of complex vector bundles on $S^8(S^4)$, there is a ...
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Evaluate Pontryagin class on Lens space $S^5/\mathbb{Z}_k=L(k;1,1,1)$

I realize this question Poincare dual PD[$A$] of a manifold $M$ and $A\in H^1(M,\mathbb{Z}_k)$ has something to do with the Steenrod problem. I want to ask a more specific question. Let $X \in H^1(S^5/...
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Degree of a line bundle using the zero locus of a holomorphic section

I am reading Kobayashi's book DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES and in the proof of its Lemma 5.7.5, he uses the fact that when $L \rightarrow M$ is a complex holomorphic line bundle ...
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Allendoerfer and Weil's generalization of Gauss-Bonnet Theorem

In Peter Petersen's Riemannian Geometry (reference, p. 98), he says that The theorem is now called the Chern-Gauss-Bonnet Theorem despite the fact that Allendoerfer and Weil were the first to prove ...
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Â-genus of a complex manifold

I am trying to understand the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), and at the end they say that since $$TM \otimes \mathbb{C} =...
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K Group Determined by Chern Classes

Why is it that the complex reduced K group of $\mathbb{CP}^2$ is determined by Chern classes $c_1$ and $c_2$? I am aware of the fact that the cohomology ring of complex Grassmannians is generated by ...
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Algebraic topology reference for someone well acquainted with algebra and differential geometry

I have spent the last two or so years making myself well rather well acquainted with the foundational aspects of differential topology/geometry. I have also spent the last year taking a two course ...
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Definition of Chern classes : classical axiom vs Grothendieck

There are two styles for the definition of the Chern classes $c_k(E)$, for a vector bundle $E\rightarrow X$, defined by the axioms: $c_0(E)=1$, $c_k(f^*E)=f^*c_k(E)$ for a continuous map $f:X\...
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Does the first Chern class generate $H^2(P_n\mathbb{C};\mathbb{Z})$?

Let $P_n\mathbb{C}$ denote the $n$-dimensional complex projective space. We define Chern classes via the Chern-Weil theory, and then I already proved that the first Chern class $-c_1(L)$ for the ...
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How to tell the tangent bundle of $S^2$ from the bundle $S^2\times$ $y-z$ plane?

Intuitively, I would like to say that the tangent bundle of $S^2$ and the bundle ($S^2\otimes \rm{y-z\ plane}$) is different. By the latter I mean a product bundle, embedding in $R^3$ it is equivalent ...
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Classify $U(1)$ bundle over $\mathbf{P}^3$, and its topological invariants [closed]

I am interested in knowing the classification of the $U(1)$ bundle over the complex projective space $\mathbf{P}^3$. This is effectively a $U(1)$ bundle over the $6$-manifold $M^6$. What are the ...
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Lens space $L^k(n) \equiv S^{2k-1}/\mathbb{Z}_n$: Stiefel-Whitney class and non/spin manifold

Define the lens space $L^k(n) \equiv S^{2k-1}/\mathbb{Z}_n$. What is the property of Stiefel-Whitney class $w_1(TM)$ and $w_2(TM)$ for $M= L^k(n)$? What is the spin or nonspin manifold property? Is ...
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Pullback of Stiefel-Whitney class along vector bundle map

Let $\xi = (\pi: E \to B)$ be an $n$-dimensional vector bundle and let $u \in H^n(E, E_0; \mathbb{F}_2)$ be the mod $2$ fundamental class of $E$. (I.e. it is the class such that it is fiberwise the ...
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calculating Euler classes

I want to understand how to compute Euler classes, what are the canonical examples of vector bundles from which i can start, and are there any books or lectures which describe how to compute Euler ...
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Second Chern class $c_2$ of $SU(n)$ bundle over $S^4$ v.s. $\pi_3(SU(n))=\mathbb{Z}$

Let us compare the four statements: Consider the $SU(2)$ fiber over the $S^4$. Such that this $SU(2) = S^3$ fiber over $S^4$ gives a second Chern class $c_2$ on the $S^4$ with $c_2=1$. Consider the $...
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First Chern class $c_1$ of $U(1)$ bundle over $S^2$ v.s. $\pi_1(S^1)=\mathbb{Z}$

Let us compare the four statements: Consider the $U(1)$ fiber over the $S^2$. Such that this $S^1$ fiber over $S^2$ gives a first Chern class $c_1$ on the $S^2$ with $c_1=1$. Consider the $S^1$ ...
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