# Questions tagged [characteristic-classes]

Cohomology classes associated to vector bundles. Includes Stiefel-Whitney classes, Chern classes, Pontryagin classes, and the Euler class.

322
questions

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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 +...

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0
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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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0
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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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### 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, ...

2
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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-...

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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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### 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 ...

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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 ...

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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 $\mathbb{G}^\times$-...

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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 $\...

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0
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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-...

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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 ...

6
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1
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### Is there a representation of $\mathrm{SU}_8/\{\pm 1\}$ that doesn't lift to a spin group?

$\newcommand{\GL}{\mathrm{GL}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\SU}{\mathrm{SU}}\newcommand{\Spin}{\mathrm{Spin}}\renewcommand{\O}{\mathrm
O}\newcommand{\R}{\mathbb
R}\newcommand\Z{\mathbb Z}$...

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### Comparison between residual intersection in Fulton's intersection theory and Aluffi's result on Milnor class

$\textbf{Question}$ I deduced that $m(A \cup B, V) = 0 $ for nonsingular variety $V$ and nonsingular hyper surfaces $A$ and $B$ whose intersection is also nonsingular. But I do not think it is true ...

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### Characteristic classes of non-linear sphere bundles

It is well known that the diffeomorphism group of there sphere $\operatorname{Diff}(S^n)$ has the homotopy type of a product $X:=O(n+1)\times \operatorname{Diff}_{\partial D^n}(D^n)$ of the orthogonal ...

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### Concrete descriptions of $S^1$-bundles over smooth manifold $Y$ underying a K3 surface

Let $Y$ be the smooth manifold underlying a K3 surface. As a manifold, $Y$ is diffeomorphic to $\{[x_0:x_1:x_2:x_3]\in\mathbb{C}P^3\colon X_0^4+x_1^4+X_2^4+X_3^4=1\}$. It is well known that $H^2(Y,\...

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### Coefficient of the top Pontryagin class in $L$-genus

The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows:
$$L_1=\frac{1}{3}p_1,$$
$$L_2=\frac{1}{45}(7p_2-p_1^2),$$
$$L_3=\frac{1}{945}(62p_3-...

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1
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### What is the Todd class *really*?

My question is about how to think about the Todd class.
Usually this is presented via Grothendieck Riemann Roch (GRR): if $X$ is a smooth projective scheme over a field $\mathbf{C}$, the chern ...

2
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### Stiefel Whitney number of a fiber bundle

I was going through this paper, and the author rights the following
The Stiefel-Whitney class of $E$ is given by $$w(E)=(1+\alpha)^{2m+1}\left\{(1+c)^{2n+1}+u_1(1+c)^{2n}+\dots+u_{2n}(1+c)+u_{2n+1}\...

2
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### First Chern form of line subbundle

Let $\pi:E\to X$ be a holomorphic vector bundle over a complex manifold. Denote by $\tilde{E}=\pi^*E\to E$ the pullback of $E$ over itself. There exists a tautological line bundle $L\subset \tilde{E}$ ...

7
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### First Pontrjagin class and generator of $\pi_3(\mathrm{SO}(d))$

It is well-known that $H^4(B\mathrm{SO}(d), \mathbb{Z}) \cong \mathbb{Z}$, with a canonical generator given by $p_1$, the first universal Pontrjagin class.
Let's assume $d\geq 5$ so that everything is ...

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### When are bundles of odd and even differential forms isomorphic?

Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ ...

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### Linear $S^{2k}$-bundles over $S^{4k}$

By the classification of Dold and Whitney, linear $S^2$-bundles over $S^4$ are classified by their first Pontryagin class $p_1$, which takes the value $4\lambda$ for the bundle corresponding to $\...

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### Are there local maps of simplicial (co-)cycles on $d$-manifolds beyond cohomology operations?

I'm interested in locally defined maps of cocycles/chains on manifolds of a fixed dimension $d$ which are compatible with cohomology. To be concrete about what "local" means, let me consider ...

2
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1
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### Preimage by birational maps

I am looking for an example (I guess that in complex projective space $\mathbb{P}^{n}$ is good) such that satisfy the following condition (in non trivial case, for this assume $X \neq \tilde{X}$):
Let ...

4
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### Multi-variable cohomology operations

Intuitively, cohomology operations are ways to locally compute a cocycle $\alpha\in H^i(X, G)$ from any cocycle $\beta\in H^j(X, H)$. Formally, they are in one-to-one correspondence with homotopy ...

3
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### What characteristic classes are there?

Can someone concisely list all characteristic classes (i.e., the cohomology classes $H^*(BX,A)$ of the corresponding classifying spaces) for the most relevant structure groups $X$ such as $O(n)$, $SO(...

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### Do we know any examples of complex surfaces where we have explicit knowledge of the Chern–Weil functions?

Let $X$ be a compact complex surface (smooth). Let $\gamma_1, \gamma_2$ denote the Chern–Weil functions. That is, if $\omega$ is a Kähler form on $X$ with volume form $\omega^2$, then $\gamma_1, \...

0
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### Define a characteristic class on a simplicial complex (non-manifold)

Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class?
(Please provide Yes or No answers, and reasonings.)
Given a fixed ...

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0
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154
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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 $\rho$ ...

4
votes

1
answer

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### Real analogue of Segre classes

Let $X$ be a manifold and $E\to X$ a complex vector bundle and let's work in $H^\bullet(X,\mathbb{Z})$. Given the total Chern class of $E$, $c(E)=1+c_1(E)+\cdots+c_n(E)$, we can define the total Segre ...

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### Rank of matrix coming from cobordism computations

In a computation of Pontryagin-numbers of certain manifolds (see the appendix of https://arxiv.org/pdf/2109.10306.pdf for more context) we came across the following elementary problem:
Consider the ...

2
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0
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### Fibering cobordant to projectivization of a vector bundle

I was going through this Stong's paper, I am stuck in the proof of the proposition 8.4 (given below)
I understand the proof till he derives the expression for the Steenrod square operation of the ...

4
votes

2
answers

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### How much do characteristic classes fail to characterize bundles?

Given a group $G$, let $E \to B$ be a principal $G$-bundle. It is
well-known that when $B$ is a nice enough topological space (e.g.
CW-complex), such a thing corresponds to a connected component of
$...

4
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1
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### Cobordism class of projectivization of a bundle

I was reading the book "Differentiable Periodic Maps" by P.E. Conner (1979). I am stuck at the following problem given at the end of section 21:
Let $\xi\to V^n$ be a $k$-plane bundle over a ...

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### Which Stiefel-Whitney numbers can be extended to manifolds with boundaries?

The Stiefel-Whitney numbers are classical topological manifold invariants obtained by integrating some local quantity (a cup product of Stiefel-Whitney classes) over the manifold. Which Stiefel-...

3
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### Justification for the definition of equivariant curvature

Let $G$ be a compact Lie group which act on a smooth manifold $M$.
Let $\mathbb{C}[\mathfrak{g}] \otimes \mathcal{A}$ be the algebra of polynomial maps from $\mathfrak{g}$ to $\mathcal{A}(M),$ we ...

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### Chern-Weil theory on some noncompact groups, and characteristic classes in differential cohomology

$\newcommand{\Z}{\mathbb Z}\newcommand{\HdR}{H_{\mathrm{dR}}} \newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\g}{\mathfrak g}$I have a specific question about invariant polynomials for some Lie groups,
...

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### LaTeXed "Lectures on characteristic classes" [closed]

I don't know if this is the right place to ask, but...
Is anyone interested in a LaTeXed version (by me) of "Lectures on characteristic classes" by Milnor in 1957?
Of course the successor &...

3
votes

1
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### Different ways of defining the Chern character of a complex

Consider a finite complex $E$ of (holomorphic) vector bundles on a (complex) manifold $X$, i.e, the complex is of the form
$$
0 \to E_N \to E_{N-1} \to \dots \to {E_0} \to 0,
$$
where the bundles are ...

3
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### Discrete spectrum of Dirac operator

It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so
that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete.
For example at least for $d=4$, this ...

7
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2
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### Even, non liftable Stiefel-Whitney class

Let $M$ be a smooth manifold and $E$ a smooth real vector bundle of even rank over $M$.
If $E$ admits of a complex vector bundle structure $\mathcal E$ ($\mathcal E_\mathbb R=E$) then all odd Stiefel-...

13
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3
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### Why do Chern classes and Stiefel-Whitney classes satisfy the "same" Whitney sum formula?

The Whitney sum formula for Stiefel-Whitney classes, $w_n(V \oplus W) = \sum w_i(V) w_{n-i}(W)$, looks a lot like the one for Chern classes $c_n(V \oplus W) = \sum c_i(V)c_{n-i}(W)$. But I don't know ...

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### Comparing the Segre classes of a cone with its abelian hull

Let $X$ be a smooth scheme, with a sheaf of graded quasi-coherent algebras $\mathcal{A}^*$, that yields a cone $C$ (in the sense of Fulton's intersection theory). Suppose that $\mathcal{A}^1$ is a ...

2
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1
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### Projectively flat connection

Let $E \to B$ be a Hermitian vector bundle. If $E$ has a projectively flat connection, then its total Chern character has the form $\mbox{ch}(E) = \mbox{rank} \cdot \exp(\mbox{slope})$. Is the ...

7
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### Whitney sum via Gysin

Let $E_1\to E\to E_2$ be a short exact sequence of vector bundles. The Whitney sum formula says that $e(E)=e(E_1)e(E_2)$, i.e. that the Euler class is multiplicative.
Is there a proof of this fact ...

7
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### Motivation for the definition of complex orientable cohomology theory

PRELIMINARY DEFINITIONS:
Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have:
$$
\tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt)
$$
So there is a special ...

5
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1
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### Curvature as infinitesimal holonomy 2

This question may be seen as a follow up of this original question. I'm learning Cheeger-Simons differential characters (reading Differential Characters of Bär and Becker).
If I understand correctly, ...

7
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1
answer

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### de Rham-invariants of a Riemannian metric

$\DeclareMathOperator{\Sym}{Sym}$For $N>0$, consider the $O_N$-representations $V = \mathbb R^N$ and $M_n = \ker (\Sym^n{V}\otimes\Sym^2 V\to \Sym^{n+1} V\otimes V)$ (the irreducible $GL_n$-...

2
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1
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### Shulman's Thesis on Characteristic Classes

I am trying to find a copy of H. Shulman's 1972 Berkeley thesis 'On Characteristic Classes'. I've seen it referenced in Bott's 'On the de Rham theory of Certain Classifying Spaces' but I can't seem to ...