# Questions tagged [characteristic-classes]

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

**6**

votes

**0**answers

118 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

212 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

185 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

74 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

138 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 ...

**6**

votes

**1**answer

191 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

361 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

194 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 $...

**9**

votes

**1**answer

142 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

213 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

287 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

118 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

110 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

94 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

83 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

122 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 \;...

**15**

votes

**0**answers

281 views

### Beyond smoothness-the clear picture about the notion of a differential form

In this paper N.Teleman constructs the signature operator on an arbitrary (closed, oriented) Lipschitz manifold with coefficients in a vector bundle $\xi$. In particular the notion of a differential $...

**17**

votes

**1**answer

868 views

### What are the possible Stiefel-Whitney numbers of a five-manifold?

On a compact five-manifold, the Stiefel-Whitney number $w_2w_3$ can be nonzero.
An example is the manifold $SU(3)/SO(3)$, and also another example is a $\mathbb{CP}^2$ bundle over a circle where the ...

**9**

votes

**1**answer

227 views

### Fourth obstruction, Pontryagin and Euler class

Assume the first three obstruction classes of a rank 4 vector bundle vanish and look at the fourth obstruction class. This fourth obstruction class can be decomposed as the Euler class and the first ...

**6**

votes

**0**answers

161 views

### $U(1)$ v.s. $SU(N)$ v.s. $SO(N)$ instantons

I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of:
Chern class (1st, 2nd), and
...

**1**

vote

**0**answers

98 views

### Conventions / Normalizations of Yang-Mills Field Theories

Let the spacetime be 4-dimensional.
In the usual Maxwell theory of Abelian gauge fields $A$, where field strength $F=dA$ one considers the Maxwell action written as
$$
S_{Maxwell}\equiv\int
-\frac{...

**1**

vote

**1**answer

57 views

### Example of a certain partitioned manifold

I'm looking for an example of a non-compact spin manifold $M$ and a compact subset $K\subseteq M$ such that $\partial K$ is a compact hypersurface in $M$ with $\hat{A}(\partial K)\neq 0$.
(At first I ...

**3**

votes

**1**answer

230 views

### Chern classes of generators of $K(S^{2n})$

Calculate the Chern classes $$ c_n \in H^{2n}(S^{2n})$$ for the generator of the group $$ K(S^{2n})$$ where $S^{2n} $ - sphere of dimension $ 2n $, $ K(S^{2n})$ - group from K-theory.
I found the ...

**3**

votes

**0**answers

115 views

### Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(2)$ or $BO(2)$

Thanks to a suggestion by @Igor Belegradek, I am interested also in a simpler problem of this earlier question 301523, by knowing what can we say about the classification of fibrations for classifying ...

**5**

votes

**1**answer

263 views

### Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(3)$ or $BO(3)$

I am interested in knowing what can we say about the classification of fibrations for classifying spaces $B^2M \equiv B^2\mathbb{Z}_2$ and $BG \equiv BSO(3)$ or $BO(3)$.
Here we can take either:
$B^...

**4**

votes

**1**answer

139 views

### Computing the Euler class of a vector bundle

I'm having the following problem: let $T \subset G := SO(2k)$ be the maximal torus acting on $V := \mathbb{R}^{2k}$ by linear transformations on each $2$-dimensional component. Denote by $V_T := (V \...

**4**

votes

**1**answer

290 views

### A question about the Chern-Weil construction of Euler class

I'm reading Appendix C of "Characteristic Classes" by Milnor & Stasheff and something confuses me. When proving that the Pfaffian of the curvature form (of an oriented $2n$-plane bundle $\xi$ over ...

**10**

votes

**0**answers

101 views

### k-th Pontryagin class of $\Lambda^{2k}_{\pm}$ on an oriented $4k$-manifold

If $M^{4k}$ is an oriented Riemannian $4k$-manifold, then the star-operator splits the bundle $\Lambda^{2k}$ into $\pm 1$-eigenspace bundles denoted $\Lambda^{2k}_{\pm}$.
I'm curious if anyone has ...

**3**

votes

**0**answers

50 views

### Will the transgression formula for superconnections give back the transgression formula of connections?

Let $E$ be a vector bundle on a smooth manifold $X$ and $\nabla$ be a connection on $E$, by Chern-Weil theory, the Chern character of $(E,\nabla)$ could be construct as
$$
ch(E,\nabla):=tr(\exp(-\...

**2**

votes

**0**answers

70 views

### Do we have a transgression formula for the chern characters of quasi-isomorphic cochain complexes of vector bundles?

Let $(E^{\cdot},d_E^{\cdot})$ be a cochain complex of complex vector bundles on a smooth compact manifold $X$. Now for each $E^i$ we could assign a connection $\nabla_E^i$ and obtain its curvature $(\...

**7**

votes

**1**answer

362 views

### First Chern class of a specific line bundle

Let $E$ be a spin$^c$ bundle and $spin^c(E)$ the corresponding $spin^c(n)$-principial bundle. Let $g_{U,V}: U \cap V \to spin^c(n)$ denote transition functions for this principial bundle and consider ...

**3**

votes

**1**answer

376 views

### How does one introduce characteristic classes [closed]

How does one introduce, or how were you introduced to characteristic classes?
You can assume that the student is comfortable with principal bundles and connections on principal bundles.
I am not ...

**13**

votes

**2**answers

491 views

### Why is the first integral Pontryagin class a homeomorphism invariant?

At the end of his 1956 paper On Manifolds Homeomorphic to the 7-Sphere, Milnor shows that either
There exists a closed topological 8-manifold with no smooth structure; or
The first Pontryagin class $...

**4**

votes

**0**answers

39 views

### Characteristic classes of invariant star products

Let $\mathfrak{g}$ a Lie algebra, can one compute the $\mathfrak{g}$-invariant Deligne class of an invariant star product by using some kind of invariant local $\nu$-Euler derivation?

**3**

votes

**1**answer

362 views

### Prerequisites for reading characteristic classes

Can some one tell me what are the prerequisites for learning characteristic classes as they are in book Foundations of Differential geometry by Kobayashi and Nomizu.
I only read first two chapters of ...

**1**

vote

**0**answers

95 views

### Extending the definition of positivity from line bundles to vector bundles

A line bundle over a complex manifold is called positive is if its Chern class is the fundamental form of a Kaehler manifold. For vector bundles of higher rank, the Chern class is no longer in general ...

**8**

votes

**0**answers

191 views

### Dixmier-Douady class is the third integral Stiefel-Whitney class

Let $M$ be (say smooth) manifold. From the short exact sequence of groups $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}_2 \to 0$ (where the first map is multiplication by $2$) one obtain long exact ...

**1**

vote

**0**answers

111 views

### Characteristic classess of Cliford bundle of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold.
Let $E$ be the Cliford bundle associated to $TM$.
Does the structure of $E$, as a vector bundle depend on choosing the Riemannian metric $g$? How can we write ...

**4**

votes

**0**answers

88 views

### tertiary characteristic class: integration of the Chern-Simons form

Let $P \to M$ be a trivial principal circle bundle with connection $A$ over a closed 3-manifold $M$. The Chern-Simons 3-form of the connection is defined by $\mathrm{CS}(A) = A \wedge dA$. Suppose ...

**10**

votes

**1**answer

406 views

### Mathematical/Physical uses of $SO(8)$ and Spin(8) triality

Triality is a relationship among three vector spaces. It describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation ...

**7**

votes

**3**answers

308 views

### Stiefel-Whitney class of an orthogonal representation

Let $BG$ denote the classifying space of a finite group $G$. For which group cohomology classes $c\in H^2(G;\mathbb{Z}/2)$ does there exist a real vector bundle $E$ over $BG$ such that $w_2(E)=c$?

**6**

votes

**0**answers

188 views

### Defining the Euler class in different ways

Let $\pi: E\to M$ be a rank two real vector bundle over a manifold $M$. Bott and Tu defines the Euler class by:
giving $M$ a Riemannian structure,
taking a trivializing chart $U_\alpha$ of $M$,
...

**10**

votes

**1**answer

338 views

### Is there a closed 5-manifold $M$ with $w_1(M)w_2(M)\ne 0$?

I'm trying to find generating manifolds for the cobordism group $\mathit{MO}_5(K(\mathbb Z/2, 2))\cong (\mathbb Z/2)^4$, which can be represented as the cobordism group of closed 5-manifolds $M$ ...

**2**

votes

**1**answer

218 views

### Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$

Do we know, or are there any References that list down complete oriented and unoriented Bordism Group $Ω_{n,O}(pt)$ and $Ω_{n,SO}(pt)$ of points $pt$ for dimensions $n=1,2,...,10$?
Here are some ...

**3**

votes

**1**answer

328 views

### A second cohomology class associated to a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold of dimension at least $4$.
We consider the differential operator $$D:\Gamma(TM)\to \Gamma (TM)$$
with $$D(X)=\nabla \circ Div(X)$$.
The principal ...

**2**

votes

**1**answer

122 views

### A line bundle on the wedge sum of spheres associated to a polynomial $P(z)\in \mathbb{C}[z]$

Assume that $P\in \mathbb{C}[z]$ is a polynomial of degree $n$ with $n$ distinct roots $z_1,z_2,\ldots,z_n$.
We identify $\mathbb{R}^3$ with $\mathbb{C}\times \mathbb{R}$. Put $a_i=(z_i,0)$.
Then ...

**12**

votes

**2**answers

742 views

### A Compact Manifold with odd Euler characteristic whose tangent bundle admits a field of lines

I understand that the top Stiefel Whitney class is an obstruction for the tangent bundle of a manifold to have a trivial line sub-bundle. I am looking for a counterexample when removing the word "...

**3**

votes

**0**answers

135 views

### Obstruction to the existence of lifting of the classifying map

Let $E$ be an $n$-plane bundle over CW complex $X$. Then $E$ is a pullback of tautological bundle $\gamma_n$ over $BO(n)$ i.e. $E=f^*(\gamma_n)$. This $f$ is called classyfing map. One can show that ...

**5**

votes

**2**answers

251 views

### Two set of axioms for Stiefel-Whitney classes

Let $E \to X$ be a vector bundle. We can associate to $E$ several invariants: among them are the Stiefel-Whitney classes $w_i(E) \in H^i(X;\mathbb{Z}_2)$. These classes may be defined using the axioms:...

**12**

votes

**2**answers

375 views

### Steenrod powers of Pontryagin classes

It is well known that the Stiefel–Whitney classes $w_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w_{2^{i}}$. I wonder if it the same statement is known/true ...