# Questions tagged [characteristic-classes]

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

253
questions

**-3**

votes

**0**answers

55 views

### Spectral sequence of classifying space of coefficient $\mathbb{Z}/2$, Bockstein sequence and integral cohomology of classifying space [migrated]

$H^*(G_{2m+1}(\mathbb{R}^\infty);\mathbb{Z}/ 2)$ forms a cochain complex with respect to the differential operator $\mathrm{Sq}^1$. Compute the cohomology.
By the Bockstein exact sequence
$$\dots \...

**9**

votes

**1**answer

464 views

### Are all classes Stiefel-Whitney classes?

When I thought of this question, I was sure it must have been asked before on this site, but I could't find anything. Maybe my search skills are lacking, or maybe the question is obvious and it's my ...

**4**

votes

**1**answer

214 views

### Action of Steenrod algebra on Chern classes

This is question about result from Brown and Peterson $H^*(MO)$ as an algebra over the Steenrod algebra. Unfortunately, the paper is not available on the Internet, so I can't find the proof.
One of ...

**16**

votes

**4**answers

987 views

### Analogy between Stiefel-Whitney and Chern classes

There is a clear similarity between Stiefel-Whitney and Chern classes, if one replaces base field $\mathbb R$ with $\mathbb C$, coefficient ring $\mathbb Z/2$ with $\mathbb Z$ and scales the grading ...

**5**

votes

**0**answers

175 views

### Derivative of the Bott-Chern forms

The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...

**3**

votes

**1**answer

146 views

### Definition of 1st degree obstruction class

Recently I go through obstruction class illustrated by Milnor.
He defined $\mathfrak{o}_i$by an element in $H^i(M; \pi_{i-1}(V_{n-i+1}(F))$, which is cohomology with local coefficients.
But the 0th ...

**2**

votes

**0**answers

118 views

### If $n$ is not a power of 2 then the dual Stiefel-Whitney class $\bar{w}_{n-1} = 0$

Stiefel-Whitney classes are invertible and for $w$, the Stiefel-Whitney class of the tangent bundle of $M$, we have its inverse $\bar{w}$. I want to prove that if $n$ is not a power of 2 then the dual ...

**6**

votes

**0**answers

130 views

### Geometric theory for cohomology groups $H^p(M;\mathbb{Z})$

An excerpt from the book Loop Spaces, Characteristic Classes and Geometric Quantization by Jean-Luc Brylinski is mentioned below:
Characteristic classes are certain cohomology classes associated
...

**1**

vote

**1**answer

83 views

### Relation between compact vertical cohomology and local cohomology groups

I'm reading the books by Bott & Tu and Milnor & Stasheef simultaneously. The following is my doubt:
The Thom isomorphism in Bott & Tu is obtained as $H_{cv}^{*+n}(E)\rightarrow H^*(M)$, ...

**5**

votes

**0**answers

110 views

### Integration on an non-orientable manifold [closed]

Suppose $M_n$ is a $n$ dimensional non-orientable manifold.
I am interesting in knowing whether the following statements are true:
A characteristic class $w_{n}^{(p)} \in H^{n}(M_n, \mathbb{Z}_p)$...

**4**

votes

**1**answer

293 views

### Bordism groups of $X$, Thom isomorphism and characteristic numbers

Recap: bordism group
An oriented singular $n$-manifold in $X$is a map $f:M^n\to X$ where $M$ is a finite disjoint union of $n$-dimensional smooth manifolds.
The empty set is an admissible oriented ...

**6**

votes

**0**answers

82 views

### Can one relate $g(\nabla_XX,\nabla_YY) = |\nabla_XY|^2$ with some characteristic class?

Let $(M^2,g)$ be a closed surface and let $X,Y\in C^{\infty}(TM)$ such that $[X,Y] = 0$. I am working on a problem where I have to deal with the following term
$$g(\nabla_XX,\nabla_YY) - |\nabla_XY|^...

**3**

votes

**1**answer

263 views

### Chern classes of complex vector bundle

I'm reading characteristic classes form the book Differential forms in Algebraic Topology by Bott and Tu. The Chern classes are defined as follows:
$E\xrightarrow{\rho} M$ is a vector bundle and $E_p$...

**4**

votes

**1**answer

192 views

### Chern -Weil map for topological principal G bundles

Let $G$ be a Lie group.
In the book Curvature and Characteristic classes, the author (Johan L. Dupont) mentiones in beginning of chapter 5 the following :
The notion of a topological principal $G$...

**11**

votes

**3**answers

528 views

### A binary operation on vector bundles that adds Chern classes?

Let $E$ and $F$ be two complex vector bundles over a space $X$. There's a fairly well-known binary operation called the direct sum, written $E\oplus F$, which has the property that its first Chern ...

**15**

votes

**0**answers

277 views

### Can the intermediate Chern classes be expressed as Euler classes?

General question: We know that the top Chern class $c_n(\xi)$ of an $n$-dimensional complex vector bundle $\xi$ is its Euler class, while the first Chern class, $c_1(\xi)$, is the Euler class of its ...

**2**

votes

**0**answers

218 views

### Splitting principle for real vector bundles

I'm reading the Book of John Roe, Elliptic Operators, Topology and Asymptotic Methods and got stuck at Lemma 2.27.
i) How does this lemma show that a real vector bundle can be given by a pullback of ...

**3**

votes

**0**answers

99 views

### About the proof of Milnor-Novikov theorem about multiplicative generators of (complex) bordism ring

I am trying to understand part of Milnor-Novikov theorem about multiplicative generators of $MU_* \cong \mathbb Z[x_1, x_2, \dots]$ using S.Kochman’s “Bordsim, Stable Homotopy and Adams Spectral ...

**4**

votes

**3**answers

265 views

### Electromagnetism as a $U(1)$-gauge theory

I would like to learn gauge theory, starting from the simplest case. I have heard that I should start with electromagnetism, which is just the $U(1)$-gauge theory. All the references I know are ...

**2**

votes

**1**answer

132 views

### Pontryagin square of first Stiefel-Whitney class

Let $w_1$ be the 1st Sitefel-Whitney classes of the tangent bundle of a 4-manifold $M$ ($M$ is non orientable). My question is
Is $$\exp\left(\frac{i\pi}{2}\int_{M_4} \mathcal{P}(w_1^2)\right)$$ ...

**4**

votes

**0**answers

91 views

### Understanding $w_2$ as an obstruction to trivializing the tangent bundle over 2-cells

I am reading through "A Geometric Proof of Rochlin's Theorem", and it is occurring to me, again, that I don't understand spin structures / $w_2$. My confusion arrises in, naturally, the proof of ...

**11**

votes

**0**answers

306 views

### Elementary-ish geometric proof of Hirzebruch signature theorem for Riemannian 4-manifolds?

The Hirzebruch signature theorem tells us that for a smooth compact oriented 4-manifold, the signature $\sigma(M)$ is proportional to the first Pontryagin number of $M$:
$$
3\sigma(M)= p_1(M) = k \...

**2**

votes

**1**answer

243 views

### Advantages of Atiyah sequence version of connections on a principal bundle

I am reading Lie Groupoids and Lie Algebroids in Differential Geometry
by Kirill Mackenzie.
In appendix (page $291$), before discussing about Atiyah sequence associated to a Principal bundle, the ...

**5**

votes

**1**answer

306 views

### Using Stiefel-Whitney class to build new principal bundles

I'm reading this paper and at the beginning of the second section, he states many results that aren't clear to me.
Consider a principal $SO(3)$-bundle $P\rightarrow R^2\times \Sigma$, where $\Sigma$ ...

**5**

votes

**1**answer

235 views

### A vector bundle associated to a codimension $1$ submanifold of a symplectic manifold

We consider the standard symplectic structure $\omega=\sum dx_i\wedge dy_i$ on $\mathbb{R}^{2n}$. To every codimension $1$ submanifold $M\subset \mathbb{R}^{2n}$ we associate a vector bundle ...

**16**

votes

**1**answer

457 views

### Wu formula for manifolds with boundary

The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=...

**4**

votes

**0**answers

284 views

### Chern-Weil theory and Weil homomorphism of principal bundle

In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $G$ bundle. In this book, they use connection on a principal bundle to ...

**8**

votes

**1**answer

274 views

### On the classification of $\mathrm{SU}(mn)/\mathbb{Z}_n$ principal bundles over 4-complexes

In
The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513
Woodward proposed a classification of $\mathrm{PU}...

**7**

votes

**0**answers

87 views

### Relate two different mod 2 indices: $\eta$ invariant and the number of zero modes of Dirac operator, associated to SU(2)

My major question in this post here is that:
How can we relate the following two mod 2 indices:
$\eta$ invariant,
the number of the zero modes of the Dirac operator $N_0'$ mod 2,
associated to ...

**1**

vote

**0**answers

51 views

### Possibility of defining Chern character form in terms of odd Chern Character form?

Given a complex vector bundle $E\to X$ with a connection $\nabla^E$ and an automorphism $U$ of $E\to X$, one can define an odd Chern character form $\textrm{ch}(\nabla, U)$ in terms of Chern character ...

**9**

votes

**0**answers

196 views

### Invariant polynomials in curvature tensor vs. characteristic classes

Let $M$ be an $4m$-dimensional Riemannian manifold. We can then form the Pontryagin classes $p_k(TM)$ of the tangent bundle using Chern-Weil theory. For any sequence of numbers $k_1, \dots, k_l$ such ...

**5**

votes

**0**answers

180 views

### Generalizing the formula between Wu class and the Steenrod square

I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy
$$
Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) .
\tag{eq.1}$$
...

**15**

votes

**3**answers

530 views

### Is the cohomology ring $H^*(BG,\mathbb{Z})$ generated by Euler classes?

I am interested in the classifying space $BG$ of a finite group $G$.
A real representation $V$ of $G$ of dimension $r$ defines a real vector bundle over $BG$ of rank $r$. If the determinant of this ...

**1**

vote

**0**answers

110 views

### Lift up characteristic class to chain complex

In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space. ...

**8**

votes

**0**answers

216 views

### Generalize Wu formula to general Bockstein homomorphisms

The classical Wu formula claims that
$$Sq^1(x_{d-1})=w_1(TM)\cup x_{d-1}$$
on a $d$-manifold $M$, where $x_{d-1}\in H^{d-1}(M,\mathbb{Z}_2)$.
I wonder whether there is a generalization of the ...

**5**

votes

**1**answer

285 views

### Conversion formula between “generalized” Stiefel-Whitney class of real vector bundles: O(n) and SO(n)

$O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$,
$$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$
Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as:
$$
w_j(...

**7**

votes

**0**answers

254 views

### Different definitions of Stiefel-Whitney classes

It is quite easy to show that different definitions of the Stiefel-Whitney classes agree by showing that they satisfy the well-known axioms. Nevertheless I have been asking myself wether one can prove ...

**5**

votes

**0**answers

89 views

### Group cohomology of “twisted” projective SU(N) with various coefficients

Given a group
$$
G= PSU(N) \rtimes \mathbb{Z}_2,
$$
where $PSU(N)$ is a projective special unitary group. Say $a \in PSU(N)$, $c \in \mathbb{Z}_2$, then
$$
c a c= a^*,
$$
which $c$ flips $a$ to its ...

**2**

votes

**0**answers

162 views

### Characteristic classes in term of cocycles

Giving a vector (principal) bundle is equivalent to give a family of cocycles ${g_{\beta \alpha}: U_\alpha\cap U_\beta \to G}$ where $G$ is the structure group of the bundle.
Chern classes are ...

**7**

votes

**1**answer

239 views

### Action of diffeomorphism group on non-vanishing vector fields

Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0)...

**8**

votes

**2**answers

395 views

### Can one disjoin any submanifold in $\mathbb R^n$ from itself by a $C^{\infty}$-small isotopy?

Let $M$ be a manifold and $V$ be an oriented vector bundle. It's well known that if the Euler class of $V$ is non zero, then $V$ can't have a non-vanishing section. The converse is not true, see ...

**5**

votes

**1**answer

248 views

### Characteristic classes of the bundle of trace free, skew adjoint endomorphisms

In "Floer Homology groups in Yang-Mills theory", Donaldson says that if we take an $U(2)$-vector bundle $E$ and we construct the bundle $\mathfrak{g}_E$ of trace-free, skew adjoint automorphisms of $...

**10**

votes

**1**answer

235 views

### Discrete Pin structures

It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of ...

**4**

votes

**0**answers

229 views

### Generalized Postnikov square

Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, ...

**4**

votes

**1**answer

315 views

### (Co)bordism invariant of Eilenberg–MacLane space becomes vanished

Consider a (co)bordism invariant
$$
u_2 Sq^1 u_2+Sq^2 Sq^1 u_2
$$
obtained from
$$
\Omega^5_{O}(K(\mathbb{Z}/2,2)).
$$
Here $u \in H^2(K(\mathbb{Z}/2,2),\mathbb{Z}_2)$. The $K(\mathbb{Z}/2,2)$ is ...

**5**

votes

**1**answer

141 views

### Pontryagin square, Postnikov square and their consistency formulas

$\mathcal{P}_2$ is Pontryagin square
$$H^{2i}(M,\mathbb Z_{2^k})\to H^{4i}(M,\mathbb{Z}_{2^{k+1}}).$$
$\mathfrak{P}$ is the Postnikov square $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$
...

**3**

votes

**1**answer

214 views

### Bockstein homomorphism and Square Operations: Their consistency formulas

Here are various ways to define "Bockstein homomorphism:"
Let $\beta_p:H^*(-,\mathbb{Z}_p)
\to H^{*+1}(-,\mathbb{Z}_p)$ be the Bockstein homomorphism associated to the extension $$\mathbb{Z}_p\to\...

**2**

votes

**0**answers

113 views

### Pontryagin square on spin and non-spin manifold

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely,
$$
\mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x.
$$
...

**2**

votes

**0**answers

85 views

### Inflation of $w_j(V_{SO(N)})$ and $w_j(M)$ from $SO(N)$ to $Spin(N)$ or Spin geometry

We know well this short exact sequence
$$
1 \to \mathbb{Z}_2 \to Spin(N) \to SO(N) \to 1.
$$
The $j$-th Stiefel-Whitney class of the associated vector bundle of $SO(N)$, as $w_j(V_{SO(N)})$, can be ...

**3**

votes

**0**answers

155 views

### Pairing the Arf with Stiefel-Whitney class

The Arf invariant is a nonsingular quadratic form over a field of characteristic 2.
The form that I looked at was:
$$
S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;...