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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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A description of the map between Grassmanians $G_1^k \rightarrow G_k$,

We know that $G_k:=co\lim G_k(\Bbb C^n)$ is the classifying space for $k$ dimensional complex vector bundles. With total space $E_k = \{(x,v) \, :|, x \in G_k, v \in \Bbb C ^\infty \}$. So we may ...
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Stiefel-Whitney numbers of manifolds that are boundaries of non-smoothable manifolds

Can a smooth compact manifold be the boundary of a non-smoothable manifold? If so can any of its Stiefel-Whitney numbers be non-zero? Thom's theorem says that a compact smooth manifold has zero ...
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Stiefel-Whitney numbers of exotic differentiable structures

The same topological manifold given two distinct differentiable structures comprises two different smooth manifolds. Do these two smooth manifolds have the same Stiefel-Whitney numbers? In other words,...
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Classification of circle bundles over $\mathbb{RP}^2$

I am trying to understand isomorphism classes of bundles $$\mathbb{S}^1\hookrightarrow E\to \mathbb{RP}^2.$$ These are classified by homotopy classes $[\mathbb{RP}^2,BAut(\mathbb{S}^1)]$. ATTEMPT 1. ...
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Chern class of tautological line bundle over the projectivization of a vector bundle

Let $\mathbb{C}^k\hookrightarrow E\to B$ be a complex vector bundle. Let $\mathbb{CP}^{k-1}\hookrightarrow\mathbb{P}(E)\to B$ be its projectivization. We can consider the tautological line bundle $L$...
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Spin structure and characteristic classes

I do not know if anyone can help me with these doubts of spin structures and characteristic classes. 1) Is there an orientable manifold that is not spin? 2) Is there a finite group $ G $ such that ...
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Chern class of fiber bundle

I have a fiber bunde $F\to E\stackrel \pi \to B$ and i want to calculate its first Chern class $c_1(E)=c_1(TE)$. How can i do this? I read here, that $$TE \stackrel \sim = \pi^* TB \oplus T_\pi E, $$...
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Is there an orientable $3$-manifold with non-vanishing $w_2$?

In the case that $M$ is a closed orientable $3$-manifold, using Wu's formula we can show $w_1(M) =0 \implies w_2(M) =0$, and so $w_3 = w_1w_2 + Sq^1 w_2 = 0$ (or you can use the fact that $\chi(M)=0$ ...
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Existence of $S^1$-action on a vector bundle and computing its characteristic classes

The existence of an $S^1$ action sometimes helps us in computing topological invariants. For example we can compute the Euler characteristic looking at the fixed point set (see Euler characteristic ...
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Spin structure and bordism

I have some questions about bordism and spin structures on manifolds. If you have any examples or references I would appreciate it. Is there a 3-manifold $ M $, orientable, which does not support 3 ...
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Dual Stiefel-Whitney class $\overline{w}_{n-1}$ vanishes if $n$ is not a power of 2

This is part of problem 11-E in Milnor and Stasheff's Characterisitic Classes, which reads: Prove the following version of Wu's formula. Let $$ \overline{Sq}:H^\Pi(M)\rightarrow H^\Pi(M) $$ be the ...
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Criteria of a real vector bundle to be Stably trivial

I am interested to know the general condition for a real vector bundle $V$ over an unorientable manifold $X$ ($X$ can be either 4d or 5d) to be stably trivial. The case I've heard of is when $X$ is ...
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Can we detect non-trivial bundles using characteristic classes for a custom structure group?

Suppose $E\to X$ is a fibre bundle where say $X$ is a CW complex, with structure group $G$, i.e. it is classified up to bundle isomorphism by a homotopy class of maps $c\colon X\to BG$. It's not ...
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Characteristic classes for $P \rightarrow G \rightarrow G/P$

Let $G$ be a complex semisimple Lie group and let $P$ be a parabolic subgroup. We know that the cohomology of the flag variety $G/P$ is generated by Schubert classes. There is a principal $P$ bundle, ...
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Chern class of holomorphic symplectic manifold

Let $M$ a complex surface and $\omega\in H^0(\Omega_M^2,M)$ a non degenerate holomomorphic form. I've read somewhere (without proof), that then the first chern class of the symplecitc manifold $(M,Re~ ...
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Isomorphism of bundles , Madsen's Calculus to Cohomology

Let $\xi=(E,M,p)$ be a smooth bundle over $M$. Denote $\Omega^0(\xi)$ the smooth sections. We also define $\Omega^i(M):= \Omega^0(\wedge^i T^*M)$, differential $i$ forms, and $\Omega^0(M)$ is then ...
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A “natural homomorphism” $H^n(B; \Bbb Z) \rightarrow H^n(B; \Bbb Z_2)$.

This is Proposition 4.12, pg33 The claim of the statement is: The natural homomorphism $\Gamma:H^n(B;\Bbb Z) \rightarrow H^n(B ; \Bbb Z_2)$ sends the Euler class to the top Stiefel Whitney class. ...
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Ring isomorphism between $H^*(G_n(\Bbb F^\infty); R)$ and Ring of characteristic classes

Definitions: Define the following map: let $k \in H^m(G_n(\Bbb F^\infty ), R)$, and $\xi= (E,B,p)$ then define $k(\xi) = g^*(k) \in H^m(B,R)$, where $g$ is the induced map from the bundle map $(F,g):\...
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Chern class of a principal $G$ bundle for a compact Lie group $G$

This question is related to this question. The user who asked this question is not active since September. So, asking a separate question here. Let $G$ be a compact Lie group and $P\rightarrow M$ be ...
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Chern class cohomology coefficients complex/real/integral?

I am reading Chern classes from Kobayashi and Nomizu. Given a vector bundle $\pi:E\rightarrow M$ with fibre $\mathbb{C}^r$ and Group $GL(r,\mathbb{C})$ they associate for each $k\leq r$ a cohomology ...
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Is there a difference of sign conventions of Dirac Index between mathematics and physics?

In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by $$\mathrm{Ind}(D\!\!\!\!/_{A})=\frac{-1}{8\pi^{2}}\int_{M}\mathrm{Tr}(F\wedge F)+\frac{...
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Todd class of a two-dimensional bundle

Let $M$ be a complex 2-dimensional manifold. I am suggested to prove that the Todd class of $TM$ is $1$, but i can't quite believe that it indeed holds. Given a bundle $\xi: E \rightarrow X$ of rank $...
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Inverse problem : Finding a vector bundle (resp. with connection), given its characteristic class (resp. differential character)

Given a rank $n$vector bundle $\alpha :E \to M$, and an element $u \in H^k(BG, \mathbb{Z})$, $G=GL(n,\mathbb{R})$we can define its characteristic class $u(\alpha) \in H^k(M, \mathbb{Z})$ as $f_\alpha^*...
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Examples of 4-manifolds with nontrivial third Stiefel-Whitney class $w_3$.

What are some examples of $4$-manifolds $M$ for which the class $w_3(TM)\in H^3(M;\mathbb{Z}/2)$ is nontrivial? Is there a mapping torus with this property? Motivation: I am wondering whether any ...
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$W$ spin implies $\partial W$ spin

Let $M$ be a compact orientable manifold with the first two Stieffel-Whitney numbers equal to zero (this is my definition of SPIN manifold). Let $B$ be the boundary of $M$; I want to show that $B$ is ...
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Euler class as obstruction to have a never vanishing cross section

We know that (see Hatcher's vector bundles and K-theory Prop. 3.22) the Euler class of an orientable vector bundle or rank $r$, $E\to M$ is the first obstruction to the existence of a never vanishing ...
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Orientation for a vector bundle

After the definition of orientation for a vector bundle$^{(1)}$ in "Characteristic Classes" by Milnor/Stasheff (page 96), the author make this comment: The local compatibility condition implies ...
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chern class of bundle in Blochs “Cycles on arithmetic schemes”

In Blochs "Cycles on arithmetic schemes and Euler characteristics of curves" he defines the bivariant chern class for bundles on a scheme. Now I wanted to calculate a bivariant chern class with $n=1$ ...
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Path components of space of almost complex structures

Let $M$ be a closed oriented smooth 4-manifold which admits an almost complex structure. The Ehresmann-Wu theorem states that a class $c\in H^2(M;\mathbb{Z})$ is realizable as the first Chern class of ...
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Extend the fiber of a principal $PSU(n)$-bundle

For $n>2$, the outer automorphism group of $PSU(n)$ is $\mathbb{Z}_2$. My question: Given a principal $PSU(n)$-bundle $P$ over a manifold $M$, can we extend the fiber of $P$ to $\mathbb{Z}_2\...
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Characteristic classes of unit sphere bundle

Let $M$ be a smooth manifold and $\xi:E\to M$ a real vector bundle over $M$. Suppose we fix a metric $g$ on $E$ so that we can define the unit sphere bundle $S(E)\to M$. How are the characteristic ...
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Fiberwise Characteristic Classes

I recently became interested in a problem of detecting the (non)existence of a section for a bundle where the fibers are not constant. Let $I$ be the unit interval and let $F_t$, $t \in [0,1]$ be ...
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Chern classes of $S^2$

It's known that $S^2$ is a $1$-dimensional complex manifold. Let $\varepsilon^n$ denote the trivial vector bundle of rank $n$, then $TS^2\oplus\varepsilon^1 = \varepsilon^3$, so by the Whitney product ...
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Stiefel Whitney classes on the simplex or the simplicial complex

The Stiefel Whitney classes of the base manifold $M$ are characteristic class as $$ w_j(M) \in H^j(M,\mathbb{Z}_2), $$ Puzzle: How do we write $$ w_1(M) \in H^1(M,\mathbb{Z}_2) $$ $$ w_2(M) \...
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two versions of the Euler class

I was wondering about cohomological and k-theoretical Euler class, or both versions of characteristic class in general. I mean, one knows that characteristic classes can measure, how twisted such a ...
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$H^1(M,\mathbb{Z}_2)$: 1st Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure

$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient, it is often to see that we say the 1st Stiefel Whitney class ...
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Godbillon-Vey class is independent from the choices involved

In the section 2.3 of these notes the Godbillon-Vey class is constructed. It is shown that this class does not depend from the choices involved (lemma 2.11). I have troubles understanding the ...
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What is the formula for the Wu class $v_6$ in terms of Stiefel-Whitney classes?

Please let me know what is the formula for the Wu class $v_6$ in terms of Stiefel-Whitney classes. Many thanks.
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$\Omega_4^{SO}(K(\mathbb{Z}_2,2))$ v.s. $H^4(K(\mathbb{Z}_2,2),U(1))$: Cocycle form

The $SO$ bordism group of Eilenberg–MacLane space $K(\mathbb{Z}_2,2)$ is $\Omega_4^{SO}(K(\mathbb{Z}_2,2))=\mathbb{Z}_4$. The cohomology group of $K(\mathbb{Z}_2,2)$ with $\mathbb{R}/\mathbb{Z}=U(1)$...
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Sum formulas for Pontrjagin square and Postnikov square

Inspire by this, I wonder Pontrjagin square: There is a geometric interpretation of $\mathfrak{P}_2$, due to Morita. Assume $q=2k$, so that the Pontrjagin square is a map $$\mathfrak{P}_2 \colon H^{...
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Kirby–Siebenmann class and the 4th Stiefel-Whitney class: $ \operatorname {ks} (M)$ v.s. $w_4(M)$

Kirby–Siebenmann class $ \operatorname {ks} (M)$ is an element of the fourth cohomology group $$ {\displaystyle \operatorname {ks} (M)\in H^{4}(M;\mathbb {Z} /2)} $$ which must vanish if a ...
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How can it be that the Chern class fully determines a line bundle, but having Chern class zero doesn't imply a line bundle is trivial?

It is well-known that the Chern class of a line bundle in $H^2(M,\mathbb Z)$ fully determines the bundle up to isomorphism. However, in this wikipedia entry on Calabi-Yau manifolds it is stated that ...
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Existence of an element in $K^0$ group, Koszul complexes

I havee such a question on construction of the Koszul complex (further we are concerned about K-theoretical Euler class). I was thinking of introducing the Koszul complex, and the existing of elements ...
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Submanifold generators that correspond to Stiefel Whitney class of $\mathbb{RP}^n$

Here let me write the real projective space $\mathbb{RP}^n \equiv RP^n$. I computed that $$w_1(RP^5)=0,$$ $$w_2(RP^5) \in H^2(RP^5, Z_2) \neq 0,$$ thus it is non-zero. $$w_3(RP^5)=0,$$ $$...
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Some questions about the choice of the definition of characteristic classes.

In the following link : http://math.uchicago.edu/~may/REU2017/REUPapers/Malaney.pdf , page : $13$ , there is a paragraph, which says : When looking at complex characteristic classes with ...
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Chern classes of tangent bundle over the Grassmannian G(2,4)

What are the Chern classes of the tangent bundle $\tau_G$ of the Grassmannian $G=G(2,4)$ of lines in $\mathbb{P}^3$? This is Exercise 5.37 on page 191 of 3264 & All That by Eisenbud and Harris. ...
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Why do the first Chern classes of these line bundles span the Dolbeault cohomology group $H^{1,1}(X;\mathbb{R})$?

Forgive me for what is probably a simple question, I am new to this field. I am studying the Hirzebruch surfaces and their higher dimensional analogues $M_{n,k}$, defined to be the projective line ...
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Virtual vector bundles, Grothendieck group, and a pair of bundles

I hope to understand the following expression better: $$w_3(V_{SO(3)}\otimes (TM-5)) = w_1^3(TM)+ w_3(V_{SO(3)}).$$ Here $V_{SO(3)}$ is the vector bundle of $SO(3)$. The $TM$ means the tangent ...
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How to pair the Arf with Stiefel-Whitney class?

The Arf invariant is a nonsingular quadratic form over a field of characteristic 2. The form that I looked at was: $$ S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;...
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unit element in cohomology and k-theory

It is some well-known fact that there is the unit element in $K^{-2}(point)$ whereas there is no unit in $H^{-2}(point)$? Where it is come from? For details I add the link http://pages.uoregon.edu/...