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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The isomorphism $H^1(X;\mathbb Z_2) \rightarrow \operatorname{Hom}(\pi_1(X),\mathbb Z_2)$ and $w_1(E)$

On page $87$ of Hatcher's book Vector Bundles and K-Theory it states that, assuming $X$ is homotopy equivalent to a CW complex ($X$ is connected), there are isomorphisms $$H^1(X;\mathbb Z_2) \...
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The $n$-th Stiefel–Whitney class $w_n \ne 0$ indicates that every section of the vector bundle (rank $n$) must vanish at some point

What I read: If $w_n \ne 0$, where $n$ is the rank of the vector bundle, then there cannot exist one everywhere linearly independent section of the vector bundle. The $w_n \ne 0$ indicates that every ...
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Does a non-trivial orientable vector bundle with rank $n$ have $n$ independent sections?

Why I know A bundle of rank $n$ is trivial iff it has $n$ linearly independent sections. We can write $B\times \mathbb R^n \rightarrow E(b,t_1,...,t_n)=\sum_i t_is_i(b)$ (where $t_i\in \mathbb R$) ...
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If the first Stiefel-Whitney class of a subset $\gamma$ of a vector bundle $E$ is $w_1(\gamma)\neq 0$, is $w_1(E)\neq 0$ as well?

Define: A vector bundle $E$ over a m-dimension base space $M$, $f: E\rightarrow M $ A non-contractible loop $L$ in the base space $M$ The vector bundle of the loop $L$ is $\gamma$ , $g: \gamma \...
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Prove the first Stiefel-Whitney class of the line bundle (as a Mobius strip) over the base space $S^1$ is $w_1=1$

What I know: A trivial vector bundle $E$ has a vanished first Stiefel-Whitney class $w_1(E)=0$. If a vector bundle $E$ is non-orientable (as a vector bundle, not a manifold), $w_1(E)\neq 0$. The ...
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Prove the non-orientability of a vector bundle using the first Stiefel-Whitney class $w_1 \neq 0$

Question: How to compute the first Stiefel-Whitney class $w_1$ of the following vector bundle? Or Prove it is $w_1\neq 0$. The base space is a torus $T^2$ The fiber is locally $\mathbb{R}$ The line ...
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The first Stiefel-Whitney class of a trivial line bundle $E=M \times \mathbb{R}$

If the base space $M$ is non-orientable, is the trivial line bundle $E=M \times \mathbb{R}$ also non-orientable? i.e. $w_1(E) \neq 0$. If so, how could it be proved? Could we use $w_k(\xi\times\eta)=\...
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Gysin -/ Shriek map of projectivized bundle

Consider the Euler sequence $$0\rightarrow \gamma \rightarrow \epsilon^4 \rightarrow Q \rightarrow 0$$ over $\mathbb{P}^3$. Take the projectivized bundle $\pi:P(Q)\rightarrow \mathbb{P}^3$ and ...
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Proving that the number of conjugacy classes of a finite group $G$ equals the number of $G$'s irreducible representations

Preamble: I want to understand why the number of irreducible representations of a finite group equals the number of conjugacy classes of the group. The only direct proof I managed to find is in the ...
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Cohomology Ring of Formal Infinite Series (Milnor)

In chapter 4 of Milnor/Stasheff's Characteristic Classes, the authors define the ring $H^\prod(B; \mathbb{Z}/2)$ to be the ring of formal infinite series: $a = a_0 + a_1 + a_2 + \cdots$ , where $a_i \...
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$SU(2)$-bundles over four-manifolds are determined by top cells

Suppose we have a compact oriented simply-connected four manifold $X$ and a $SU(2)$ bundle $P$ with $2$nd Chern class $=k$ over $X$. We know $X$ have a top cell $e^4$, if we collapse all lower cells ...
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Misunderstanding of Chern character

I'm trying to understand the basics of the chern character before I get much more involved. I'm really confused since (3) and (5) in the picture below seem to conflict. One says that addition goes to ...
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$K$-theory of $S^2$: spinor bundle vs tautological bundle over $\mathbb{C}P^1$

I'm trying to understand the relationship between different generators of the $K$-theory group of $S^2$. Part of my curiosity comes from reading this discussion about characteristic classes. The $K$-...
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Isomorphism class of vector bundle over $S^2$ and the Chern character

Let $V$ be a complex vector bundle over $S^2$. Suppose that The rank of $V$ is $1$; $\int_{S^2}\text{ch}(V)=1$. I've seen it written that these two facts determine $V$ up to isomorphism. Question 1: ...
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Divisor of a meromorphic section of a line bundle represents the first chern class

The following is taken from p.279 of Scorpan's The Wild World of 4-Manifolds. For a meromorphic section $f$ of a holomorphic line bundle $L$ over a complex surface, the linear combination of curves $\...
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Is the first Stiefel-Whitney class an isomorphism if there is a unique orientable class?

Suppose that $X$ is a nice compact manifold such that its reduced real $K$-group $\tilde{K}\mathcal{O}(X)$ has a unique stably equivalent class of orientable bundles, i.e, $\ker(\omega_{1})$ is the ...
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Stably Equivalent and Isomorphism Class of Real Line Bundles

Let us denote by $\text{Vect}_{1}(X)$ the set of isomorphism classes of real line bundles of a compact nice manifold $X$. It is well known that the reduced $K$-group $\tilde{K}\mathcal{O}(X)$ can be ...
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Triviality of complexified tangent bundle of spheres

What is known on the triviality of the complexified tangent bundle of the spheres? Are they always trivial? Here is my progress. If we are just looking at any rank n complex vector bundle, then this ...
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One unclear detail in Baum 1968 paper "On the cohomology of homogeneous spaces"

I was asking the same kind of question about a year ago, haven't got any distinct answer then, afterwards I got distracted, and now when I'm thinking about that question again - I still can't come up ...
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Either $\pm c_1(S)$ is ample or $c_1(S)=0$ for a normal projective surface with quotient singularities with $b_2(S)=1$

According to this paper: https://arxiv.org/pdf/math/0602562.pdf, in p.2 (below Theorem 1), it is written that if $S$ is a normal projective surface (so there are only finitely many isolated ...
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reference about slant product

I'm recently study the orientation of ASD moduli space, which involved some techniques about slant product in algebraic topology. However I never heard about this kind of product before. So I wonder ...
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Characteristic Classes of Fiber Bundles of Closed, Aspherical Manifolds with Isomorphic but Non-Congruent Group Extensions

Suppose I have two closed, aspherical manifolds, $F$ and $B$, with $K = \pi_1(F)$ and $Q = \pi_1(B)$, and two group extensions $1 \to K \to G_1 \to Q \to 1$ and $1 \to K \to G_2 \to Q \to 1$ with $G_1$...
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Definition of the Polynomial $P(D, E)$ in Riemann-Roch Without Denominators.

In the proof of Lemma 15.3 in Fulton's Intersection Theory, there appears the formula $$c(\Lambda^\bullet D^\vee \otimes E) = \prod_{p=0}^d \prod_{j=1}^e \prod_{i_1 < \dotsb < i_p} (1 + y_j - ...
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Well-defineness of Chern roots?

Let $\mathcal{E} \to X$ be a rank $n$ (complex) vector bundle on a space $X$ (possibly with some other mild conditions to make the splitting principle holds). According to the splitting principle, ...
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Relation among Chern classes

Let $E\rightarrow\mathbb{P}^2$ be a rank $r$ vector bundle over $\mathbb{P}^2$ with Chern classes $c_i = c_i(E)$. Is there any relation among $c_1$ and $c_2$ or are they in general completely ...
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How can the Chern-Simons form trivialize a non-trivial characteristic class?

I have a very basic confusion about Chern-Simons forms: On Wikipedia and other sources, it is stated that the Chern-Simons 3-form $$Tr(A\wedge dA+\frac23 A\wedge [A\wedge A])$$ trivializes $Tr(F^2)$, ...
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5 votes
1 answer
96 views

Triviality of complexified tangent bundle

Let be $M$ be a smooth manifold (possibly with boundary) and assume that the complexified tangent bundle $TM \otimes \mathbb{C}$ is trivial. Does this imply that $TM$ is stably trivial? This seems to ...
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4 votes
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Easy computation by Ravi Vakil (jet bundles)

I landed on some short notes by Ravi Vakil from the 90s, the Beginner's Guide to Jet Bundles from the Point of View of Algebraic Geometry. The notes are very clear but on the very first page there is ...
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6 votes
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Confused about notation for the cohomology of Grassmannians

In Milnor and Stasheff's book "Characteristic Classes", problem 7b, they ask us to show that the cohomology ring $H^* (G_n (\mathbb{R}^{n+k}, \mathbb{Z}_2)$ is generated by the Stiefel-...
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Algebraic Independence of Miller-Morita-Mumford classes of surface bundles (following Morita)

I am trying to understand the proof of algebraic independence of the characteristic classes of surface bundles, as outlined in Morita's book "Geometry of characteristic classes" (Theorem 4....
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Integral of the $n$-th power of the first Chern class of the hyperplane bundle

Let $\mathbb{P}^n$ denote the complex $n$-dimensional projective space, also let $\mathcal{O}(1)$ denote the hyperplane bundle, i.e. the dual bundle of the tautological bundle $\mathcal{O}(-1)$. I ...
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Finite coverings and Tangent bundles

Motivation: Apparently the tangent bundle of real projective space is trivial if and only if the tangent bundle of it's universal cover (the sphere) is trivial. That is, $ n=1,3,7 $. Does that follow ...
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1 vote
1 answer
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The Stiefel-Whitney classes of a tangent bundle as a manifold

For any smooth manifold $M$, the tangent bundle $TM$ as a manifold is always orientable. In other words, the first Stiefel-Whitney class $w_1$ of the manifold $TM$ always vanishes. Question: Does the ...
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Manifolds with zero Chern term

Has any study been made of manifolds where the integrand $e(\Omega)$ from the Chern-Gauss-Bonnet theorem (which I gather is called the Euler class) is identically zero? If so, is there any broader ...
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4 votes
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Calculation about Chern character in a special setting

I'm confused with working out the Chern character in the following special setting. Let $E$ be a spinor bundle $$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$ over sphere $S^{2n}$, where $\rho$ ...
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Stiefel–Whitney class and Pontryagin class requirement on a simplicial complex: triangulation and branching structure

Given a simplicial complex with only triangulation and branching structure, is it enough to define Stiefel–Whitney class and Pontryagin class? If so, could you explain how to obtain these ...
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Chern classes of quaternionic projective plane

We begin with $\mathbb{H}^2$ and we let $[q_1,q_2]$ denote the class $\{ (q q_1, q q_2), q \in \mathbb{H}, q \neq 0\}$. Under stereographic projection, there is a biejction between this set and $S^4$. ...
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2 votes
1 answer
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How to calculate the Euler class, Euler characteristic and top Chern class of $End(E)$?

It's me again. Could someone please ilustrate the relationships between these concepts through the following example: Let $E$ be a rank 2 holomorphic vector bundle on $\mathbb{CP}^2$. Find the Euler ...
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3 votes
2 answers
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A formula for the second Chern class of the tensor product of a line bundle and vector bundle

If possible I would like someone to prove or suggest a place to see the proof of this relation: $$c_2(V \otimes L)=c_2(V)+(r−1)c_1(V)c_1(L)+ {r \choose 2} c_1(L)^2$$ Here $L$ is the line bundle and $r$...
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Clarification with Pontryagin classes of the 4-dimensional sphere $S^4$

In some homework I am asked the following: what are the Pontryagin classes of the 4 dimensional sphere $S^4$. My doubt is: should I assume that I am actually being asked about the Pontryagin classes ...
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3 votes
1 answer
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Chern classes and sums of line bundles

Let $E$ be a complex vector bundle of rank $r$ and suppose we can write $E = \oplus_{i=1}^r L_i$ where $L_i$ are line bundles. I have read here (and think I more or less understand why) that the total ...
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2 votes
1 answer
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Is the canonical $(n-1)$-plane bundle over the deleted total space of an orientable $n$-plane bundle orientable?

Let $\omega : E \to B$ be a real $n$-plane bundle, and $E_0$ the subspace of $E$ containing all the non-zero vectors. We can construct an $(n-1)$-plane bundle on $E_0$, where the fiber above a vector $...
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4 votes
1 answer
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Recognising Chern-Weil forms

Given a smooth vectorbundle $E\to B$ with connection $\nabla$, the (real or complex) characteristic classes of $E$ are the cohomology classes of the Chern-Weil forms associated to $\nabla$. Suppose $E$...
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Theta characteristic and spin structures

In many places one can find the statement that the existence of of a Spin-structure on a (compact) Kähler manifold $M$ of (complex) dimension $n$ is equivalent to the existence of a $\theta$-...
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1 vote
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Example of a real orientable $2n$-plane bundle without complex structure via non-trivial odd Stiefel-Whitney class

For any complex vector bundle, the odd Stiefel-Whitney classes of its underlying real vector bundle are trivial. So if a real vector bundle has a non-trivial odd Stiefel-Whitney class, it is not ...
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First Chern class of the cotangent bundle vanishes

I'm interested in the first Chern class of the cotangent bundle. I concretely work on the sphere $S^2$, but the reasoning below seems to work for any manifold. I take the symplectic point of view ...
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Classification of sphere bundles and vector bundles over a surface

The general question for which I want an answer is: Given $n\geq3$ and a closed surface $S$ of genus $g\geq1$, what are all the rank $n$ real vector bundles over $S$ (up to isomorphism)? What are all ...
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2 votes
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Chern class and Euler class

We know that for a complex vector bundle $E$ and curvature $F_{\nabla}$ on it.Then we can define the top Chern class as $det(\frac{i}{2\pi}F_{\nabla})$. Then we view $E$ as a even dimension real ...
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7 votes
1 answer
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What is the action of Steenrod squares on $BSO(n)$?

My specific question is as follows: In $H^\ast(BSO(3);\mathbb{Z}/2\mathbb{Z})\cong \mathbb{Z}/2\mathbb{Z}[w_2,w_3]$, what is $Sq^2(w_3)$? I seem to have proven (below) that the answer is ...
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7 votes
0 answers
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Axioms and uniqueness for the Euler class

In this question it was asked if the 4 properties listed on the wikipedia page uniquely characterise the Euler class. I answered no and claimed: For every oriented vector bundle $E\to X$ of rank $n$ ...
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