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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Differentiable way of detecting characteristic classes of odd (real) dimension vector bundles? [closed]
I learned that Chern classes work for complex vector bundles (corresponding to a real bundle of even dimensions). Equivalently, Pontryagin classes work for even dimensions.
Is there a characteristic ...
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characteristic classes in odd dimensions? [closed]
So Stiefel-Whitney, Pontryagin, Chern classes are all for even dimensions. There seem to be no characteristic classes for odd dimensions. Does that mean in odd dimensions everything is trivial?
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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
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 possible ...
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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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Proof of local triviality of the vector bundle $\gamma_n^{1}$ for the base space $\mathbb{P}^n$
Milnor defines the total space of $\gamma_n^1$ as follows
Let $E\left(\gamma_n^1\right)$ be the subset of $P^n \times R^{n+1}$ consisting of all pairs $(\{\pm x\}, v)$ such that the vector $v$ is a ...
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About naturality of Godbillon-vey class [closed]
This is a problem from Lawrence Conlon's differential manifolds a first course. I do not know how to prove in the following problem
If $f: N \rightarrow M$ is transverse to $\mathcal{F}$, prove that
$$...
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Vanishing first Chern class implies existence of flat connection
Edit: In the exercise I tried to solve it wasn't stated explicitly that we were dealing with line bundles (though the usage of the symbol $L$ for the vector bundle was a strong suggestion that this ...
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How to calculate the first Chern class (concrete simple example)?
I have been finding a good source on computing first Chern class of a manifold (e.g.Riemannian) but I just couldn't find a single example of how to compute in terms of local coordinates. I wanted to ...
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Can all Chern numbers be interpreted as the degree of some maps?
In some physics applications, I am aware that the first Chern number and the second can be interpreted as the degree of some maps.
For example, the first Chern number appears in topological insulators,...
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Real Line Bundle Corresponding to a Double Cover
I am studying spin structures on $SO(n)$-bundles using some lecture notes. Right after defining the twist of a spin structure $(P,\psi,\rho)$ on $Q\xrightarrow{} X$ by a double cover $\pi:R\...
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Euler number of symmetric cube of the tautological bundle
Let $E$ -is a tautological two-dimensional bundle (rank $n=2$) over complex Grassmannian $\operatorname{Gr}(2, 4)$ ($2$-dimensional planes in $C^4$). I'm trying to compute the Euler number $\oint_{\...
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Ambiguity in trace of curvature 2-form matrix
My question is about computing the first Chern class from the trace of curvature 2-form matrix via the Chern–Weil theory. First Chern class is given by
$$ c_{1}=\left[{\frac {i}{2\pi }}\operatorname {...
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How the k-th Chern class looks like?
I have a question about the definition of the Chern class. I have studied the subject via the introduction of invariant polynomials ("Loring W.Tu Differential Geometry").
Let $P(X)$ a ...
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Question on top Chern class of ample vector bundle
In W. Fulton, R. K. Lazarsfeld - Positive polyonomials for vector bundles, Ann. of Math. 118 (1983) 35-60, they prove the positivities of the top Chern class of a rank $r$ ample vector bundle $E$ over ...
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Ideal sheaf of $Z$ twisted by $Z$
Let $X$ be a nonsingular projective surface and $Z$ a 1-dimensional subscheme (i.e. a curve) and $m:= c_2(I_Z)$.
What (geometrically) is $m$? What is the intuition behind it?
What is $I_Z(Z)$? In ...
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Extend the tangent bundle on a modular curve to its compactified curve, with a Hermitian metric singular at cusps
The brief setting of the problem is the following. Suppose we are given the Poincare metric on the open modular curve, and the metric is singular at cusps. The Chern form of the tangent bundle defined ...
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Chern classes of tensor product of rank two vector bundles
This answer gives the first three Chern classes of the tensor product of any two locally-free sheaves.
I computed the fourth Chern class as
$$
c_4(E\otimes F) = \frac{1}{2} c_1^4(E) -5c_1^3(E)c_1(F) -\...
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What is needed to define the index of a vector field?
All of the definitions that I have seen for the index of a vector field $X$ on $\mathbb{R}^n$ at some singularity $p$ go as follows: first let $D$ be a contractible domain on which $p$ is the only ...
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First Chern class of zero locus of a pairing
Let $E\to X$ and $F\to Y$ be vector bundles over (sufficiently nice) surfaces or 3-folds.
If I know that a divisor $D \subset X\times Y$ is the zero locus of a pairing $\alpha\colon E \otimes F \to \...
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Are complex vector bundles on an Riemann surface admitting flat connections trivial？
I wonder if the following statement is true. Or under what kind of condition this could be true?
Suppose $M$ is a compact Riemann surface. $E$ is a complex vector bundle on $M$ admitting a flat ...
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Isomorphic vector bundles have same characteristic classes
Following is an excerpt from the book "Differential Geometry: Connections, Curvature, and Characteristic Classes" that defines naturality of characteristic classes.
My question is how does ...
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Properties of image of sub-vector bundle
I have the following problem:
Let $E,F \rightarrow C$ vector bundles over $C$, with $C$ an algebraic curve of genus $g>1$ and $rank(E)=rank(F)+1$, let $ p: E \rightarrow F$ a surjective vector ...
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Proving Chern forms are closed via Bianchi identity
Let $M$ be a smooth real manifold and $E \rightarrow M$ an Hermitean vector bundle over it. Define Chern classes as
$$c(M)=\sum_{i=0}^m c_i(E)t^i=\det \left( \textrm{Id} +\frac{i}{2\pi}F \right) = \...
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Does a twisted tangent bundle $T\mathbb P^n \otimes \mathcal O(d-1)$ ever have a globally nonvanishing section?
Let $X = \mathbb P^n$ be the usual projective space over an algebraically closed field. For what values of $d \in \mathbb Z$ does the twisted tangent bundle $E = TX \otimes \mathcal O_X(d-1)$ have a ...
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First Stiefel-Whitney class of a line bundle over non-orientable surface
Let $\Sigma_g$ be the orientable surface of genus $g$ and $N_{g+1}$ be the non-orientable surface of genus $g+1$. Assume that $\Sigma_g$ is embedded inside $\mathbb{R}^{3}$ such that it admits the ...
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relations between flat metric and flat modules
The definition of flat metric has two definitions: 1. given a metric norm $F$ on manifold $M$, there exists coordinate charts s.t. for every point $p$, all differentials of the norm is zero, i.e. $\...
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homotopy equivalence between graph and sphere and related Euler characteristic
I saw a question here The relationship between the Euler characteristic and the fundamental group of finite connected graphs saying that :
Any connected graph is homotopy equivalent to wedge of $S_1$
...
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Reference request: positive first chern class of tangent bundle implies anticanonical line bundle is ample
I am searching for a reference (preferably with a proof) for the following result:
Let $X$ be a smooth projective curve, $T_X$ its tangent bundle, $K_X$ its canonical bundle. If $c_1(T_X) > 0$, ...
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Euler characteristic of the complex torus
In his book An Introduction to Invariants and Moduli, Mukai claims (Chapter 12.2) that
$$\chi(X,L) = \frac{(c_1(L))^n}{n!}$$ for any line bundle $L$ on the complex torus $X$. Now, it is clear that $...
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A difficult integral for the Chern number
The integral
$$
I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +(m-\cos x-\cos y)^2\right)^{3/2}}
$$
gives the ...
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How did Thom calculate $MSO(k)$ using Silber's polyhedron in his 1954's paper?
I am now reading Thom's famous paper Quelques propriétés globales des variétés différentiables. In page 48, Thom used an auxiliary space $K$, which is a principal fiber bundle with base space $K(\...
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Is there a relation obeyed by the 2nd universal Chern class, analogous to the 2nd universal Stiefel-Whitney class?
The Stiefel-Whitney classes.
1.1: There exist $w_j \in H^j(BO(n),\mathbb{Z}_2)$ for $j\in\{1,2,\ldots,n\}$
such that $H^*(BO(n),\mathbb{Z}_2) = \mathbb{Z}_2[w_1,w_2,\ldots,w_n]$.
1.2: The map $BO(n) ...
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How to prove Poincaré-Hopf in the non-orientable case?
I understand the following proof of the Poincaré-Hopf theoerem in the case of orientable manifolds:
Proposition: Let $X$ be an oriented compact smooth manifold (without boundary).
Let $v$ be a smooth ...
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Obstruction for existence of section of projective space bundle obtained from vector bundle?
For $V \rightarrow X$ a rank-$(n+1)$ real vector bundle, let $Z_V \subset V$ denote (the image of) its zero section. Let $E_V = (V\setminus Z_V)\,/\,(\mathbb{R}\setminus\{0\})$
be the quotient of $V$ ...
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Interpretation of the obstruction class in terms of cells (Milnor-Stasheff 12.1)
In section 12 of Milnor-Stasheff's characteristic classes, they mentioned an obstruction class for the associated Stiefel manifold bundle $V_k(\xi)$ over $B$ as follows:
Definition: The associated ...
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Every representative of first Chern class defines a prinicipal connection
I put all the definitions I am using at the bottom.
We consider a principal U(1)-bundle $\pi: E \rightarrow M$ with Lie-algebra valued connection $A \in \Omega^1(E, \mathfrak{g}) = \Omega^1(E, \...
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Do Steenrod Squares have naturality with homomorphisms that don't come from continous maps
I was reading about the Steenrod Squaring operations in Milnor and Stasheff's, Characteristic Classes, now there is an axiom regarding naturality that says that given a continous map $f:(X,Y)\...
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Uniqueness of $\hat{\nabla}:C^{\infty}(\tau^{*}_\mathbb{C} \otimes \zeta) \to C^{\infty}(\Lambda^2 \tau^{*}_\mathbb{C} \otimes \zeta)$
I have a doubt in this lemma from the book Characteristic classes by Milnor and Stasheff.
Note that $C^{\infty}(\tau^{*}_\mathbb{C} \otimes \zeta) \cong C^{\infty}(\tau^{*}_\mathbb{C}) \otimes_{C^{\...
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Characteristic classes in $H^2(\; ;\mathbb{C})$ for complex line bundles. (Milnor and Stasheff, Charateristic Classes, page 306).
I am reading the proof of Gauss-Bonnet Theorem from the Characteristic Classes by Milnor and Stasheff.
There is a statement in the proof which says that:
"But the only characteristic class in $H^...
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Constructing Chern vs. C-F Chern classes
In Characteristic Classes p. 157, Milnor and Stasheff construct the Chern classes. Let $\xi:E \to B$ be a complex vector bundle of dimension $n$. We proceed inductively:
The top Chern class $c_n$ is ...
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Chern classes of a ruled surface
I am reading Petitjean's thesis (1995). I have trouble making sense of the following:
Let $\mathcal{E}$ be a vector bundle (locally free sheaf) of rank $2$ over a curve $C$ with genus $g$ and let $Z = ...
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Definition of smooth manifolds in Milnor's characteristic classes
I just started reading Milnor's characteristic classes, but I got a bit annoyed with his definition of manifolds, smooth maps, tangent spaces etc. since he embeds everything into Euclidean spaces. Is ...
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Is there a good set of lectures or notes on topology for studying characteristic classes?
I was watching the lectures of Dr. Tadashi Tokieda on "topology and geometry", which was pretty amazing that I finished all the the lectures in 3 days. What particularly helpful was the ...
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About Chern classes of symmetric powers
I have the following question
I'm training in computation of Chern classes. Let $\xi$, $\eta$ are complex vector bundles of rank two ($r=2$)
I'm trying to find $c_1(S^2 \xi \otimes \eta)$, $c_2(S^2 \...
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Relation between characteristic classes and fundamental forms of a surface
I have an embedded oriented surface $M\subset\mathbb{R}^3$. Is there a way to express it's Euler/Pontryagin classes using first and second fundamental forms of $M$? I would really appreciate a ...
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