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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Arbitrariness in the definition of characteristic classes

There are many characteristic classes which go by many names but all of them seem to be a function of the curvature two-form $F$ $$f(\mathcal{F})$$ where $f$ has a power-series of the form $$f(x) = 1+...
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Verifying a given cohomology class is the Stiefel-Whitney class

We know that Stiefel-Whitney class can be determined uniquely using the axioms, thus if we are given an cohomology class assigned to each real vector bundle, we shall be able to use the axioms to ...
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Integration over invariant polynomials

Can we define an integration over invariant polynomials? I have only seen people doing integrations of various terms in the polynomial but not the polynomial itself, so just asked. (The invariant ...
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About the obstruction theory and homotopy?

I am trying to read Hatcher's vector bundle and $K$-theory. I am trying to understand the obstruction theory in Chapter $3$. When we get a vector bundle $\pi:E\rightarrow B$ and we may suppose that $B$...
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Find Pontrjagin number

Good time of day. I try to find Pontrjagin number for such task $\langle p_1(r\gamma_H^1), [S^4] \rangle=\oint_{S^4}{p_1(r\gamma_H^1)}$ where $\gamma_H^1=(E\rightarrow \Bbb HP^1 \simeq S^4)$ - ...
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Prove that $\int_{\mathbb{C}\mathbb{P}^3}c_1(\mathbb{C}\mathbb{P}^3)^3=64$.

I want to prove that $$\int_{\mathbb{C}\mathbb{P}^3}c_1(\mathbb{C}\mathbb{P}^3)^3=64,$$ where $c_1$ is the first Chern class. I know that for projective spaces of dimension $n$ (but maybe also in ...
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class conjugate [duplicate]

I have this group $$G= \left\{ A = (a_1 ,a_2,a_3) \in GL_3(\mathbb{C}) :a_i \in \left\{\pm \pmatrix{1 \\ 0\\ 0} , \pm \pmatrix{0 \\ 1\\ 0} , \pm \pmatrix{0 \\ 0\\ 1} \right\} \right\}.$$ I know ...
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Fiber bundle of real projective space $RP^d$ over $S^1$

(1) Does there exist and (2) how many do they exist, of a nontrivial fiber bundle $N^{d+1}$ of real projective space $RP^d$ over $S^1$? $N^{d+1}= S^1 \ltimes RP^d$ so that $$ RP^d \hookrightarrow N^{d+...
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Inclusion of Finite Grassmannian in Infinite Grassmannian

I am doing Milnor-Stasheff's exercise 6-B. which is as follows: Show that the restriction homomorphism $i ∗ : H^p (G_n(R ^∞)) → H^p (G_n(R^ {n+k}))$ is an isomorphism for $p<k$. I was thinking of ...
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Vanishing Chern classes on vector bundle of $S^2$

Suppose $E\to S^2$ be a complex vector bundle. If $c_1(E)=0$, does it imply that $E$ is a trivial bundle? And why if so? This question is motivated from Audin, Damian: Morse Theory and Floer Homology
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Milnor's definition of the Stiefel Whitney number

In Milnor's characteristic class, he defines the Stiefel-Whitney class as follows: Let $M$ be a closed possibly disconnected smoooth $n$-manifold. Using mod $2$ coefficients there is a unique ...
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Is there a relationship between the multiplicity of an index and the “algebraic” multiplicity of a zero of a section from a (complex) vector bundle?

I'm a physicist who is trying to make sense of the relationship between the number of zeros of a section from an associated vector bundle and the Euler characteristic. My interest lies in applications ...
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Conflicting definitions of instanton number

In Nakahara's "Geometry, Topology and Physics" (and many other sources) the instanton number of an $SU(2)$ instanton $A$ with curvature $F^A$ is defined by $$\int_{S^4}\text{ch}_2(E)=\frac{1}...
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How to define the Stiefel-Whitney class of a complex orthogonal representation?

Background: One of the main objects of interest in the theory of $L$-functions is the root number, a complex number of modulus one which appears in the functional equation. In general, a root number ...
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Complement of zero section homotopy equivalent to $\mathrm{Gr}_{k-1}(\mathbb{C}^{\infty})$

Let $\mathrm{Gr}_{k}(\mathbb{C}^{\infty})$ be the Grassmanian of complex $k$-dimensional subspaces of $\mathbb{C}^{\infty}$, and let $\gamma:E\to \mathrm{Gr}_{k}(\mathbb{C}^{\infty})$ be the ...
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Properties of Euler Class

I am reading Milnor & Stasheff's Characteristic classes (page 98 to be precise). After defining the Euler class, they state two basic properties claiming that the proofs are obvious: 1.(...
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Orientation of a Vector Space and the Homology of a pair

I am stuck at the following argument in the book Charcteristic Classes by Milnor & Stasheff page 96: A choice of orientation for V corresponds to a choice of one of the two possible generators ...
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Does every vector bundle have a 'tensor inverse'?

For any vector bundle $E$ over a finite-dimensional CW complex, there is a vector bundle $E'$ such that $E\oplus E'$ is trivial. For a compact Hausdorff base, this is Proposition 1.4 of Hatcher's ...
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Chern character of the restriction of a canonical bundle over product space

I am new to StackExchange, and self-taught in the field of characteristic classes, vector bundles, etc... so apologies in advance if my question is somewhat trivial or ill-posed. I am trying to do a ...
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Proving Wu's Formula for Steenrod Squares (Milnor & Stasheff, Problem 8-A)

I am stuck at the following problem from chapter 8 of Characteristic Classes by Milnor & Stasheff. Problem 8-A. It follows from 7.1 that the cohomology classes $\operatorname{Sq}^kw_m(\xi)$ can ...
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Pulling Back product of Characteristic Classes

I am reading Milnor & Stasheff's Characteristic Classes but I'm having difficulty understanding the following argument from the verification of Axiom 3 in Chapter 8. $$w(\xi\times\xi') = w(\xi)\...
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Tangent bundle of the connected sum of two smooth manifolds

Let $M,N$ be two smooth connected real $n$-dimensional manifolds. Suppose they are both orientable for simplicity. Let $i:D_1\hookrightarrow M$ and $j:D_2\hookrightarrow N$ two embedded disks so that $...
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Cohomology ring of complex grassmanian as a quotient

This is about the complex version of this question, I am interested in understanding the integral cohomology ring of $\mathrm{Gr}_k(\mathbb{C}^n)$ as the following quotient of the integral cohomology ...
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Stiefel--Whitney classes of associated bundles

I would like to be able to compute the Stiefel--Whitney class of an associated bundle: if $V$ is a representation of $G$ and $P\to M$ is a principal $G$-bundle, then $$\frac{P\times V}{G}\to M$$ ...
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Cohomology of a smoothly embedded space curve

Let $\mathbb{P}^3 = P(\mathbb{C}^4)$ and $\gamma:C\rightarrow \mathbb{P}^3$ be a smoothly embedded algebraic space curve. Then its total Chern class is $c(C) = c(TC) = 1+a\in H^*(C)$, where $a^2=0$. ...
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Second Stiefel-Whitney class of $\mathbb{C}\text{P}^2\#\overline{\mathbb{C}\text{P}^2}$

I know that $w_2(\mathbb{C}\text{P}^2\#\overline{\mathbb{C}\text{P}^2})\neq 0$, where $\overline{\mathbb{C}\text{P}^2}$ is $\mathbb{C}\text{P}^2$ with opposite orientation. But how do you prove this? ...
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Orientability vs $n$-frames via Stiefel-Whitney classes

I am trying to understand Stiefel-Whitney classes as obstructions for the existence of linearly independent sections of a vector bundle $E \to B$, that is a section of the Stiefel bundle $V_k (E) \to ...
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Equation relating Chern-Simons and Bott-Chern secondary characteristic forms

Let $\overline{\mathcal E}: 0\to \overline S\to \overline E\to \overline Q\to 0$ be a short exact sequence of Hermitian vector bundles over a complex manifold $X$. Here $\overline E=(E,h^E)$ is a ...
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Equivariant Chern classes and local coefficients

I am trying to understand the basics of equivariant cohomology in view of applications to the field of crystalline topological insulators. At stake in that field is the very explicit situation of the ...
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A question about the complex conjugate bundle

If a complex vector bundle is constructed by the complexification of a real vector bundle, say $E=F\otimes \mathbb{C}$, then there's a conclusion that $E$ is isomorphic to its conjugate bundle by ...
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orientations of a manifold and its tangent bundle

Milnor's characteristic classes: Suppose M is an orientable manifold. An orientation for M is a function which assigns to each point x of M a preferred generator $u_x$ for the infinite cyclic group $ ...
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Stiefel-Whitney classes and immersion of a manifold

From Milnor and Stasheff: according to the Whitney duality theorem, $r_M$ is the tangent bundle of a manifold in Euclidean space and $v$ is the normal bundle, then the Stiefel-Whitney class satisfy: $$...
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Chern classes and curvature of total space

Let $L$ be a holomorphic line bundle on a compact complex manifold $X$ with $\dim_{\mathbb C} X = n$. The first Chern class $c_1 (L)$ can be directly related to the curvature of a connection on $L$ by ...
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Reference request for Miller-Morita-Mumford classes.

I am looking for a reference to learn about Miller-Morita-Mumford classes. I have experience in characteristic classes of vector bundles e.g. Stiefel-Whitney classes, Chern classes, Pontryagin classes,...
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Stiefel-Whitney numbers for disjoint union of a Manifold

I am reading the book by Milnor & Stasheff on Characteristics Classes, In the 4th Chapter There's is this beautiful result by Thom" If all of the Stiefel-Whitney numbers of M are zero, then M ...
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The Ricci form and the Chern class?

Let's take the tangent bundle $TM\rightarrow M$(not to be Kahler), and the first Chern class $$c_1(M)=[tr(\frac{\sqrt -1}{2\pi}\Omega)]$$ We know that the inside trace of the curvature form is a ...
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Why “Stable” in Stable vector bundle and Stable homotopy theory?

Why/what/how does stable mean in the Stable vector bundle? Stable homotopy theory? Of course, it is described in the Wiki texts in a formal way. But I wonder what are the intuitions and concepts ...
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What do characteristic classes of spinor bundle depend on?

Let $M^n$ be a smooth manifold. Equip $M$ with a Riemannian metric and let $S$ be a spinor bundle. We can consider characteristic classes of $S$ (or $S_+,S_-$ for when $n$ is even), for example the ...
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What is the difference between the first Chern class and integral first Chern class?

I'm reading a paper which says that the first Chern class of a manifold is $0$, but the integral first Chern class is not $0$. What is the difference between the two? Does it have to do with taking ...
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The three-dimensional bundle underlying a quaternionic line bundle

Consider a principal $Sp(1)$-bundle $P$ over a compact space $X$, i.e. a quaternionic line bundle. We can push $P$ forward along the double cover $Sp(1) \cong SU(2) \to SO(3)$ to get a three-...
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Euler form of a non-metric connection

Given an oriented real smooth vector bundle, the Euler class assigns a differential form on the base to each metric and metric-compatible connection on the bundle. Is there a reason that it is not ...
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Computing the euler number of an oriented disc bundle over a sphere

Suppose $E\to S^2$ is an (smooth) oriented disk bundle. Let $D_1$ and $D_2$ denote the upper and lower hemispheres of $S^2$, respectively. Then $E|_{D_1}$ and $E|_{D_2}$ are trivial, so they are both ...
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Splitting of the tangent bundle and Euler characteristic of surfaces

Let $M$ be a be a closed orientable surface, and suppose that its tangent bundle $TM$ splits into a direct sum of line bundles. How to prove that $M$'s Euler characteristic is zero? Unfortunately, I ...
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Obstruction to a Spin structure on a bundle ξ, and ξ ⊕ $n$ det ξ

In Ref, it says that: The obstruction to putting a Spin structure on a bundle $ξ (= Rn → E → B)$ is $w_2(ξ) \in H^2(B;Z/2Z)$. Pin± structures is that Pin− structures on ξ correspond to Spin ...
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Bott&Tu, Definition of the Euler class of a vector bundle

I have a question while reading chapter 11 of Bott&Tu, Differential Forms in Algebraic Topology. The book first defines the Euler class of an oriented sphere bundle (a fiber bundle with fiber $S^n$...
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Can we find a constructible sheaf whose characteristic class is the given lagrangian cycle.

I must apologize in advance for the mathematical gaps in setting up the notation and for being unable to provide a copy of the literature pertaining to this question. I have the following questions : ...
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The Euler class of any oriented sphere bundle would vanish?

I am reading chapter 11 of Differential Forms in Algebraic Topology by Bott & Tu. The argument in the following paragraph seems to imply that the Euler class of any oriented sphere bundle would ...
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How to construct a embedding from $\mathbb{R}{\text P}^{2n+1}$ to $\mathbb{R}^{4n+1}$

Whitney's embedding theorem states that any smooth $n$-manifold $M$ can be smoothly embedded into $\Bbb R^{2n}$. And we know $\mathbb{R} {\text P}^{2n+1} $ is a $2n+1$-manifold, it can be embedded ...
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Holomorphic line bundles with trivial Chern class are flat

Let $X$ be a complex, projective algebraic variety and let's work in the differential-complex setting. Let $L$ be a non-trivial hermitian holomorphic line bundle and assume that $c_1(L)=0$. Can we ...
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Equivalent definitions Stiefel - Whitney / Chern - classes

There are some details I still don't understand about the definition of Stiefel - Whitney / Chern - classes. Let $\gamma_n^1$ be the tautological line bundle over the real or complex projective space $...

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