Questions tagged [characteristic-classes]
Cohomology classes associated to vector bundles. Includes Stiefel-Whitney classes, Chern classes, Pontryagin classes, and the Euler class.
360 questions
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The first Pontrjagin class of Milnor's example of exotic 7-sphere
In the construction of exotic 7-spheres, Milnor used the fact the first Pontrjagin class $p_1(\xi_{ij})$ is linear in $i,j$. He claims that this is clear but I don't feel that. I tried to find some ...
- 151
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0
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163
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Minimal CW complex detecting all powers of euler class
Let $n \in \mathbb{Z}_{>0}$. I'm wondering about pairs $(X,V)$ where $V$ is a $2n$-dimensional vector bundle on the CW complex $X$, such that
The ring homomorphism $\mathbb{Q}[e] \to H^*(X;\...
- 869
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1
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Classification of real rank two vector bundles on nonorientable surfaces
As far as I understand from this post,
the Stiefel-Whitney classes and Euler
classes of a rank two real vector bundle on an orientable compact surface $S$
of genus $g$ characterise the bundle up to ...
- 20.2k
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96
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Natural renormalizations of K theory classes on open varieties
I have a question about natural assignments of $K$-theory classes to open varieties. The question has turned out rather long, for which I apologize.
All varieties I consider are smooth over $k=\mathbf{...
- 241
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1
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$n$-manifolds that have exactly $n-1$ independent global tangent section but is orientable?
Does there exist $n$-manifolds (possibly with boundary) that have exactly $n-1$ linearly independent global tangent section (i.e. fails to be parallelizable by 1 rank) but is orientable? orientability ...
- 189
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Computing the Pontryagin invariant of a map $f: T^3\to S^2$ with an integral formula
Given a map $g: S^3\to S^2$, we can easily compute the (integer-valued) Hopf invariant of $g$ using the Whitehead formula as follows. Let $\Omega$ be a volume form on $S^2$ normalised as $\int_{S^2}\...
- 273
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Evaluate $A p_1$ with Pontryagin Class on a 5-manifold
For the first Pontryagin Class $p_1$ that can be evaluated on a closed 4-manifold, are these true:
Lemma 1. $$\int_{M^5} A p_1 = 0 \mod 3,$$ when $A$ is $\mathbb{Z}/3$ valued 1-cochain, for a closed ...
- 10.8k
8
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219
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Are all characteristic classes of $G$-bundles captured by finite subgroups?
Let $G$ be a connected Lie group. As a general broad question I'm interested in understanding if all characteristic classes of $G$-bundles are captured by flat bundles, or some of them need curved ...
- 820
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168
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Integrating the square of the first Chern form on an almost complex $4$-manifold
Let $M$ be a smooth compact $2n$-dimensional manifold and with an almost complex structure so that the Chern classes of the tangent bundle $TM$ are defined. Choosing a Hermitian fiber metric $h$ on $...
- 427
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Grothendieck’s description of Chern classes as they appear in Hartshorne
I’m a bit puzzled as to why Hartshorne’s formula for a defining relation for Chern classes is $$ \sum_{i=0}^r (-1)^i \pi^*(c_i(\mathcal{E})).\xi^{r-i}=0 $$ (with the alternating factor of $(-1)^i$), ...
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Signature theorem for $4$-manifolds with boundary
Let $X$ be a compact oriented $4k$-dimensional Riemannian manifold with nonempty boundary $\partial X=Y$ which is isometric to a product near the boundary. The signature theorem for manifolds with ...
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A question about the "convergence" of Cheeger-Simons differential characters
Here is a question about the "convergence" of Cheeger-Simons differential characters. Let me give some background.
Let $M$ be a compact manifold. Denote by
$$\delta_1:\widehat{H}^k(M; \...
- 1,273
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Equivalent definitions of volume of representations (or characteristic classes of flat bundles)
Statement of the problem:
Let $M$ be a closed connected oriented manifold with fundamental group $\Gamma$ (considered as covering transformations acting on its universal cover). Let $G$ be a (...
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Explicit classification of real rank-2 vector bundles over compact surfaces via twisted Euler classes
There is an apparently 'well-known' complete classification of real rank-2 vector bundles (or equivalently, of principal $O(2)$-bundles) over a paracompact space $B$ in terms of characteristic classes....
- 73
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2
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Chern classes in Čech cohomology
Let $X$ be a nice topological space with a finite open covering $\{U_i\}$. Let $V$ be a complex vector bundle over $X$.
Let us fix its trivialization over each $U_i$ and construct a matrix valued 1-...
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3
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1
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Two equivalent definitions of MMM-classes
I have some trouble proving the following general result from Galatius' lecture notes on the proof of the Madsen-Weiss Theorem, slightly rephrased:
Lemma 2.3 Let $\pi:E\to X$ be a surface bundle with ...
- 325
6
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1
answer
381
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Alternating sign in a formula for the Euler class
I have been reading Peter Gilkey's Invariance Theory, the Heat Equation, and the
Atiyah-Singer Index Theorem (a pdf is available on the author's website). Overall it has been very clear so far, ...
- 193
7
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1
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Flat bundles on hyperbolic manifolds
Let $G$ be a discrete and torsion-free subgroup of $\mathrm{O}^+(n,1)$ and let $\rho\colon G \to \mathrm O(k)$ be an orthogonal representation. As I understand, we can construct a flat vector bundle $\...
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Orientability of $6$-manifold and vanishing of every Euler class
Let $M$ be a connected, closed, smooth $6$-manifold such that the Euler class of every orientable vector bundle over $M$ vanishes. What can be said about the orientability of $M$?
- 1,201
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Classification of principal $\mathrm{SO}(3)$-bundles on a 4-manifold via characteristic classes
I am interested in a reference with a detailed (as simple and topological as possible) proof of the following fact:
Theorem. A principal $\mathrm{SO}(3)$-bundle on a compact oriented 4-manifold are ...
- 8,050
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1
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Chern Classes of $\mathcal{O}_E(1)$ on $\mathbb{P}(E)$ for $E = \mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$
Let $E =\mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$ and denote by $\mathcal{O}_E(1)$ the dual of the tautological bundle.
How can I compute $c_1^2(\mathcal{O}_E(1)), c_1^3(\mathcal{O}_E(1))$, $...
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Example of principal $G$-bundle $E\xrightarrow{p}B$ with classes in the kernel of $p^{*}$ that are not characteristic classes
If $E\xrightarrow{p}B$ is a principal $G$-bundle classified by $B\xrightarrow{f}BG$, then the image of any characteristic class of $B$ under $p^*$ is trivial by the naturality of $H^*$ and the fact ...
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The discriminant of a holomorphic vector bundle
Let $M$ be a complex manifold and $E$ a holomorphic vector bundle over $M$. The discriminant $\Delta(E)$ of $E$ is then defined to be $$\Delta(E)=c_2(\text{End}(E))=2rc_1(E)-(r-1)c^2_1(E).$$
This ...
- 131
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1
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309
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Definition for the Chern–Weil formula?
I'm reading Yang–Mills connections and Einstein–Hermitian metrics by Itoh and Nakajima. On definition 1.8 they define a notion for an Einstein–Hermitian connection $A$ by $$K_A = \lambda(E)\mathrm{id}...
- 103
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1
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600
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Steenrod powers of the Thom class
René Thom in 1952 proved the formula
$$
Sq^i(U_2)=\Phi_2(w_i),
$$
which in modern parlance says that the Steenrod squares of the mod $2$ Thom class of an orthogonal bundle are the images under the mod ...
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Compute the Euler class of tautological $C$-bundle over $CP^1$
$\DeclareMathOperator\SO{SO}$This might be an old question. But since I have not found an explicit answer to this question, I put the question here.
The background is that we need to use a similar ...
- 21
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Fixed point formula of Atiyah and Singer applied to a Dirac operator on a spin manifold
Let $G$ be a compact Lie group acting by orientation-preserving isometries on a compact even-dimensional spin manifold $X$, and assume that the $G$-action preserves the spin structure of $X$, so that ...
- 427
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1
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478
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Fivebrane bordism $\Omega_d^{\mathrm{Fivebrane}}$
$\newcommand{\Fr}{\mathrm{Fr}}\newcommand{\Fivebrane}{\mathrm{Fivebrane}}\newcommand{\String}{\mathrm{String}}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$What ...
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Reference request: cohomology of BTOP with mod $2^m$ coefficients
I am searching for a reference with information pertaining to the $\mathbb{Z}/{2^m}$ cohomology of ${\rm{BTOP}}(n)$, for $n \geq 8$ and $m=1,2$, where
$${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
- 381
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Equivalence of two definitions of the $\hat{A}$-genus form
Let $E$ be a real vector bundle and $\nabla$ a covariant derivative with curvature of $F$. On page 51 of Heat Kernels and Dirac Operators it is claimed that "using the formula $\det A = \exp\...
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Possible Euler characteristics of manifolds with tangential structures
Let $p:B\to BO$ be a fibration. We say that a manifold has a $B$-structure if its stable tangent bundle lifts to $B$. I am interested in the question of whether there exists, for a given even ...
7
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1
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Why does Bott's obstruction theorem imply the vanishing of some cohomology classes of $B\Gamma_q$?
Recall that Bott's obstruction for integrability [Bott70] asserts that:
Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every ...
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222
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An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$
It is known that the only spheres that admit an almost complex structures are $\Bbb S^2$ and $\Bbb S^6$ (Borel and Serre, 1953). In particular, $\Bbb S^4$ cannot be given an almost complex structure (...
- 1,201
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Minimal first Pontryagin class $p_1=1$?
From Hirzbuch theorem,
the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$.
I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$.
Is ...
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1
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Induced fiber sequence and Eilenberg–MacLane space in Whitehead tower of $BO$
In Whitehead tower of $BO$, there is a induced fiber sequence:
1.
$$
Z_2 \to B SO \to BO \overset{w_1}{\rightarrow} B Z_2
$$
How does this map $\overset{w_1}{\rightarrow}$ from $BO$ to $B Z_2$?
...
- 467
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Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
Namely, how do we know
$$
K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)?
$$
Naively -- in each step ...
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Odd integral Stiefel–Whitney classes in terms of even ones
As computed in various places (e.g. in Brown - The Cohomology of $B\mathrm{SO}_n$ and $B\mathrm O_n$ with Integer Coefficients), the integral cohomology ring of $B\mathrm{O}(n)$ (and similarly $B\...
- 10.2k
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1
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Chern character of a super-connection (Heat kernels and Dirac operators)
Let $A$ be a super-connection on a super-bundle $E\to M$, then the differential form
\begin{equation}
\mathrm{ch}(A)=\mathrm{Str}(e^{-A^2})
\end{equation}
is called the chern character of $A$ on page ...
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1
answer
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A difficult integral for the Chern number
Cross post from Maths stack exchange
The integral
$$
I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y\phantom{,} \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +...
- 273
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$\hat{A}$-genus of a complex manifold
I am trying to understand the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), and at the end they say that since $$TM \otimes \mathbb{C} =...
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Proving a Result About Pontryagin Numbers Without Forms
I've been reading the book Geometry of Differential Forms by Shigeyuki Morita, and I came across the following theorem on page 226 the other day:
Proposition 5.53 (Pontryagin). Two cobordant closed (...
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103
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Tautological ring for moduli of flat connections
Let $X$ be a smooth complex manifold and $G$ a connected complex algebraic group. Let $M$ denote the moduli stack of flat $G$-connections on $X$. Over $M\times X$, we have the tautological $G$-bundle, ...
- 3,003
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Necessary and sufficient conditions for pseudo Riemannian manifold to be time orientable
It is well known that a smooth manifold $M$ is orientable if the first Stiefel-Whitney class of the tangent bundle vanishes. In particular, this implies that if $M$ is equipped with a pseudo-...
- 705
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160
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On Brylinski–McLaughlin's paper "Čech cocycles for characteristic classes"
In the paper "Čech cocycles for characteristic classes", the authors Brylinski and McLaughlin describe how to construct Čech cocycles with values in the Deligne smooth complex representing ...
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150
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On the positivity of the second Segre class of ample vector bundles
Let $E$ be an ample rank $r\geq2$ vector bundle over a smooth projective surface $X$ defined on an algebraically closed field $\mathbb{K}$ of characteristic $0$.
In Kleiman S. L. - Ample Vector ...
- 990
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3
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1k
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Computation on characteristic classes
I am organizing a reading seminar on characteristic classes. The audience in the seminar is interested in symplectic and contact manifolds. I work in categorification and would like to compute some ...
- 341
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226
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Bundles vs. line bundles
Let $K$ be an algebraically closed field and consider the category $\text{Bun}$ of (finite dimensional) vector bundles over a $K$-variety $X$. Consider also the category of $\mathbb{G}^\times$-...
user30211
7
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1
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645
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Meaning of the first Chern class of the unit tangent bundle of a surface
(This is a fairly basic question, not really research level, except that I am a research mathematician working on other things who is trying to understand more topology for use in my own work.)
Let $\...
- 121
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How to see $\delta_2(\hat{\chi}(V))=\chi(V)$ in differential cohomology?
I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-...
- 9,297
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3
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Are Chern classes well defined up to contractible choice?
The Chern classes are, by definition, cohomology classes. And
cocycle representatives of the Chern classes are not unique.
But it might be the case that cocycle representatives of the Chern classes ...
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