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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Show these two axioms of Stiefel-Whitney class are equivalent.

Here is a theorem about Stiefel-Whitney class in my teacher's notes: For a dim-$k$ real vector bundles $E$ over $B, k \geq 0,$ there are characteristic classes $w_{i}(E) \in H^{i}(B, \mathbb{Z}/2\...
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Computing a Gromov-Witten invariant

Some background that is not necessary for answering the question: Let $X = \mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ be a threefold. This is a $\mathbb{P}^1$-bundle ...
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Topological K-theory and characteristic classes of module bundles

Let $R$ be a commutative ring and let $A$ be a commutative unital topological $R$-algebra. By means of replacing vector spaces with $A$-modules, one can define $A$-module bundle in analogy to vector ...
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Real $2n$-plane bundle with a complex structure is a complex $n$-plane bundle

I am trying to show that if $\xi=(E,B,\pi)$ is a real $2n$-plane bundle with a complex structure $J:E\to E$ then $\xi$ becomes a complex $n$-plane bundle if we define $(x+iy)v=xv+yJ(v)$ on each fiber. ...
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Chern-Weil theory in the cohomological Atiyah-Singer theorem

I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer. Let $D:\...
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Homotopy fiber of the second Stiefel-Whitney class

The second Stiefel-Whitney class $w_2$ can be identified with a (homotopy class of) map $BSO\to B^2\mathbb Z_2$ (I write $\mathbb Z_2$ the cyclic group of order 2, and omit the dimension assumed to be ...
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Connection with torsion

Consider a metric connection $\Gamma^{\mu}_{~~\nu\lambda}$ with torsion $$\Gamma^{\mu}_{~~\nu\lambda} = \tilde{\Gamma}^{\mu}_{~~\nu\lambda}+ K^{\mu}_{~~~\nu\lambda}$$ where $\tilde{\Gamma}^{\mu}_{~~\...
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Euler class as a total derivative

In physics the Euler class of 4D manifolds is called the Gauss-Bonnet term. The Gauss-Bonnet term is called topological because it can be expressed as a total derivative. But to show this fact the ...
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Euler class of a manifold

While computing the Euler class of the manifold, can we use connections other than the Levi-Civita connection ? What is the restriction on the connection that can be used to compute the Euler class ?
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Prove addition law from duplication formula (for power series associated to elliptic genus)

I would like to prove the following statement, which I'll state initially without context since I believe it to be purely algebraic. Let $f(x)=x+a_3x^3+a_5x^5+\cdots$ be an odd formal power series (...
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Characteristic classes, Möbius strip, and the cylinder

I have been thinking how to distinguish the (open) Möbius strip from a(n open) cylinder. What does not work Standard invariants from general topology, as connectedness or compactness, Invariants ...
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Exponential of mixed-type End-valued differential form

Let $E\rightarrow \mathbb{P}^1$ be a complex vector bundle and let $a_{(0,0)},a_{(1,0)},a_{(0,1)},a_{(1,1)}$ be differential forms such that $a_{(i,j)}\in\Omega^{i,j}(\mathbb{P}^1,End(E))$. I would ...
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Errors in my calculation of Euler characteristics of $\#\Bbb{T}^{2n}$.

I want to find out the euler characteristics of the connected sum of $2n$-dimensional tori. Since we know that the euler characteristic of any odd-dimensional closed manifolds is $0$. My Attempt: ...
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Poincaré duality maps the Thom class of the normal bundle to $(-1)^{nk}i_*(\mu_M)$

This is the problem 11-C from the book, "characteristic classes" written by J.W. Milnor. Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow A$. Let $k ...
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Chern class in Bott-Tu

I've met some trouble in understanding Chern class. I first touch the Chern class in classyfying space of characteristic class. For $\pi:E\rightarrow X$ and we have such relationship:$$Vect^n_{\Bbb C}(...
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Spectral sequence of classifying space of coefficient $\mathbb{Z}/2$, Bockstein sequence and integral cohomology of classifying space

$H^*(G_{2m+1}(\mathbb{R}^\infty);\mathbb{Z}/ 2)$ forms a cochain complex with respect to the differential operator $\mathrm{Sq}^1$. Compute the cohomology. By the Bockstein exact sequence $$\dots \...
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The subspace $M=\{([x],v)\in E:|v|\leq 1\}$ of the total space of the canonical line bundle over $\Bbb RP^1$ is a Mobius band.

Let $E=\{([x],v)\in \Bbb RP^1\times \Bbb R^2: v\in [x]\}$ be the total space of the canonical line bundle over $\Bbb RP^1$. (Here $[x]$ denotes the line passing through the origin and $x\in \Bbb R^2-\{...
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Which integral Steifel-Whitney classes are universally $0$?

Let $BO(n)$ denote the classifying space of the orthogonal group $O(n)$. Then there is the well-known ring isomorphism $$H^*(BO(n);\mathbb{Z}/2) \cong \mathbb{Z}/2[w_1,\dots,w_n] $$ where $w_i \in H^...
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Splitting principle and Chern roots

I have recently seen the following definition of Chern classes, relying on classifying spaces. Let $BG$ be the classifying space of compact Lie group $G$. For the $n$-torus we have: $$H^{**}(BT^n) = ...
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Is the Poincaré dual of the first Stiefel-Whitney class of a manifold necessarily orientable?

I think the answer should be yes, and I think there's an argument for it for triangulable (compact) manifolds as follows: For our $n$-manifold $M$, given a triangulation pick some $(n-1)$-simplices ...
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The set of of all isomorphisms from one finite dimensional vector space to another has a natural topology

In p.32 of Milnor's Characteristic Classes, Milnor defines a "continuous" functor from the category $\mathfrak{V}$ consisting of all finite dimensional vector spaces and all isomorphisms between such ...
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Induction Proof for Computing Cohomology Ring of the Finite Grassmannian

I'm working on problem 7B of Milnor/Stasheff: Show that the cohomology algebra $H^*\left (G_n\left (\mathbb{R}^{n+k}\right ) \right )$ over $\mathbb{Z}/2$ is generated by the Stiefel-Whitney ...
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Naturality axiom for Stiefel-Whitney Classes

In Milnor and Stasheff's Characteristic Classes the "Naturality" axiom for Stiefel-Whitney classes is defined as follows: If $f : B(\xi) \to B(\eta)$ is covered by a bundle map from $\xi$ to $\eta$ ...
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(lectures) Reference request for Vector bundles and characteristic classes

Are there some available online lectures for first year graduate course on vector bundles and characteristic classes?
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Piecewise linear spheres

Define a piecewise linear sphere to be a simplicial complex that admits a subdivision, which is simplicially isomorphic to a subdivision of the boundary of a simplex. What's the current state of the ...
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Compute the obstruction classes of the tangent bundle of $\mathbb{R}\mathrm{P}^2$

I want to compute the obstruction classes of the tangent bundle of $\mathbb{R}\mathrm{P}^2$. I learnt obstruction classes mainly from the books on characteristic classes by Milnor. The 2nd ...
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Euler class of tautological line bundle over $\mathbf{HP}^n$.

Let $e$ denote the Euler class of the tautological line bundle $\gamma^n$ over $\mathbf{HP}^n$. My question is how to determine the pairing $$ \langle e,\ [\mathbf{HP}^n]\rangle$$ with $[\mathbf{HP}^n]...
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About theorem 16.2 in Switzer's Algebraic Topology

I have some difficulties understanding a very precise point of Switzer's proof of the existence and unicity of chern classes, which is Theorem 16.2 in his book. Unfortunatly there are many notations ...
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Chern Character of the dual bundle

Let $(E,\nabla)$ be a complex vector bundle with connection. Is there a formula for the Chern character of the dual vector bundle $(E^*,\nabla^*)$ in terms of the chern character/classes of $E$?
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Recommended books for reading the Euler characteristic class

The book Differential Forms in Algebraic Topology of Bott & Tu gives a nice treatment with the Thom isomorphisms and the Euler classes of vector bundles rank $2$. However, I think their approach ...
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Error in Milnor's Characteristic Class Chapter 12?

I've read the proof of theorem 12.5, and in the last step, he said ... If the dimension $n$ is odd, then the Euler class itself has order $2$ by 9.4, so we have proved that $\mathfrak{o}_n(\xi) = e(\...
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Bordism invariants as integrals of Stiefel-Whitney classes

I am trying to understand this mathematical physics paper by A. Kapustin, which assumes knowledge of bordism invariants of smooth compact manifolds: https://arxiv.org/abs/1403.1467v3 For example, ...
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Visualizing $|\mathcal{B}\mathbb{Z}| \simeq S^1$.

The classifying space of the integer group $\mathbb{Z}$ can be defined as the geometric realization of the underlying groupoid $\mathcal{B}\mathbb{Z}$. To unwind, $\mathcal{B}\mathbb{Z}$ is simply ...
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$S^2$ Tangent Bundle Stably Trivial?

I know that the tangent bundle $TS^2$ is stably trivial: if $\nu$ denotes the normal bundle of the embedding of $S^2$ in $\mathbb{R}^3$ as the unit sphere, then $\nu$ is a trivial line bundle. But ...
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Relation between Pontryagin classes and Chern classes, confusion with proof

I am trying to prove the following from Milnor and Stasheff but haven't been able to crack it. The hint seems to suggest that this can be done completely using the total chern classes but I haven't ...
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Is there a higher dimensional analogue to this proposition?

In Geiges' book An Introduction to Contact Topology, there is the following proposition: Proposition 4.3.2: For any even element $e \in H^2(M,\mathbb{Z})$ there is a contact structure $\xi$ on $M$ ...
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Symmetric function theorem for $A[[x_1, \dots, x_n]]^\text{symm}$

The Fundamental Theorem of Symmetric Polynomials states that if $A$ is commutative and unital, then there is an isomorphism $$A[y_1, y_2, \dots, y_n] \to A[x_1, x_2, \dots, x_n]^\text{symm}$$ given ...
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How can I evaluate the universal Stiefel-Whitney class on a given simplex?

Let $\gamma:EO(n)\to BO(n)$ denote the universal n-plane bundle and $w_i(\gamma)$ the universal Stiefel-Whitney class. Since $w_i(\gamma)$ is an element of $H^i(BO(n),\mathbb{Z}/2\mathbb{Z})$, it ...
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Stiefel-Whitney Class of the tautological line bundle over $\mathbb{CP}^1$

I am trying to compute the Stiefel-Whitney Class of the tautological line bundle, $\gamma$, over $\mathbb{CP}^1$. Since $H^1(\mathbb{CP}^1,\mathbb Z/2)$ and $H^3(\mathbb{CP}^1,\mathbb Z/2)$ are ...
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Top Stiefel-Whitney class derived from division algebra

I’m reading the proof of Bott-Milnor-Kervaire 1,2,4,8 theorem and get stuck on the following process: Why does the $w_n(\xi)$ not vanish? Or is there another way to derive the 1,2,4,8 theorem from ...
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4th Stiefel whitney class of a 7-dimensional Spin manifold

In Massey's paper "On the Stiefel Whitney classes of a manifold I" he shows that manifolds of dimension n = 4s + 3 have $w_n = w_{n-1} = w_{n-2} = 0$. Where $w_i$ is a mod 2 Stiefel-Whitney class ...
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Index of Dirac operator and Chern character of symmetric product twisting bundle

I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text (see image below). We are twisting the spinor bundle $\Sigma$ with an ...
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Seeing $\mathbb{S}^3$ as a pullback

$\require{AMScd}$ Using the Hopf Fibration $$ \mathbb{S}^1 \hookrightarrow \mathbb{S}^3 \rightarrow \mathbb{S}^2 $$ and the fibration $$\mathbb{S}^1 \hookrightarrow \mathbb{S}^\infty \rightarrow \...
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Part two of exercise 11-C of Milnor-Stasheff book

Can someone help with part two of exercise 11-C, showing that the euler class is two times a generator of $H^n(S^n)$, I think it is wrong because the Thom isomorphism should send generators to ...
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Definition and properties of a connection 1-form for the Euler (Pfaffian) curvature 2-form

Given a real and oriented $2n$-dimensional vector bundle $\pi: E \to B$, it is known that both the [$\mathfrak{so}(2n)$-valued] connection 1-form $\mathcal{A}$ and curvature 2-form $\mathcal{F} = d\...
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The definition of $s_I$ in Milnor and Stasheff, Characteristic classes, page 188

I cannot understand the definition of polynomials $s_I$ in the Milnor and Stasheff, Characteristic classes, page $188.$ In Milnor and Stasheff, Characteristic classes page $188$, the polynomials $...
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Chern character of the exterior powers

I try to understand an answer to question about Chern character of the exterior powers (Quick question: Chern classes of Sym, Wedge, Hom, and Tensor). I have 2 questions: 1) Why $ch(\Lambda(L))=1+te^{...
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Computing Chern Number of $\mathbb{CP}^1$ Tautological Bundle

I'd like to compute the Chern number of the tautological bundle of $\mathbb{CP}^1$. Consider $L \subset \mathbb{C}^2\times \mathbb{CP}^1$ given by $$ L \;\; =\;\; \left \{(w_1, w_2, [z_1, z_2]) \; | ...
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Twisted Euler class for non-orientable bundle

If $E \to X$ is an oriented rank $k$ vector bundle, then $E$ has an Euler class $e(E) \in H^k(X; \mathbb{Z})$ which is the first obstruction to the existence of a nowhere-zero section of $E$. If $E$ ...
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Chern classes of a torsion sheaf supported in dimension $0$.

Suppose that $X$ is a smooth algebraic K3-surface, and let $0 \neq \mathcal{F} \in \operatorname{Coh}(X)$ be a sheaf supported in dimension $0$. Let $\omega \in \operatorname{Pic}(X) \otimes \mathbb{R}...

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