Questions tagged [characteristic-classes]
Cohomology classes associated to vector bundles. Includes Stiefel-Whitney classes, Chern classes, Pontryagin classes, and the Euler class.
333
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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 ...
- 312
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0
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150
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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 ...
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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 ...
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3
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556
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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 $w_2(E)=c$?
- 1,189
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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 $M$,
...
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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 $M$ ...
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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 ...
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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 ...
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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 ...
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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 "...
- 953
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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 ...
- 9,150
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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:...
- 9,150
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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 ...
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Wu relation for Steenrod square and Stiefel-Whitney cocycles
Given a simplicial complex with a branching structure, we can compute the Stiefel-Whitney cocycles $w_n$.
(see R. Z. Goldstein and E. C. Turner, Proc. Amer. Math.
Soc. 58, 339 (1976))
Wu relation for ...
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Explicit cobordism between Wu manifold and Dold manifold P(1,2)?
The Wu manifold $SU(3)/SO(3)$ and the Dold manifold $P(1,2)$, the latter being defined as $(S^1\times \mathbb{C}P^2) / (p, x) \sim (-p, \overline{x})$, are cobordant because they are both generators ...
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Submanifold generator of Pontryagin-dual of the torsion subgroup of the $Pin^+$ bordism group (dimension 4th)
I am interested in figuring out the following submanifold generator of a $Pin^+$ Bordism or Cobordism group in the dimension 4, say for a $Pin^+$ cobordism group of the classifying space $BG=BSU(2)$:
...
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Is a 4-dimensional submanifold of a spin manifold always spin?
Let $M^d$ be a $d$-dimensional orientable spin manifold, and $N^4$ is a closed $4$-dimensional orientable submanifold of $M^d$.
Is $N^4$ always spin?
If $d=5$, is $N^4$ always spin?
If $N^4$ is a ...
- 4,716
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Generalize Wu formula to integral cohomology classes
For $\mathbb{Z}_2$ cohomology classes, we have a very useful Wu formula:
In $d$-dimensional manifold and for a $n$-cocycle in $x_n \in H^n(M^d; \mathbb{Z}_2)$, we have $Sq^{d-n}(x_n)=u_{d-n}\cup x_n$, ...
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Is there a four-manifold whose tangent bundle is an endomorphism bundle?
Is there a smooth four-manifold $M$ such that $TM \cong \operatorname{End}(E)$ for some rank $2$ bundle $E \to M$?
If $M$ is parallelisable, then one can take $E$ to be the trivial rank $2$ bundle ...
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Stiefel-Whitney total class with prescribed zeros
First things first, I am aware of the existence of this topic. It's related, but old and my question hasn't been discussed there. So I hope it's not wrong to start a new topic.
I'm currently ...
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What does positivity of the first Pontryagin number of a vector bundle tell us?
Some context:
In the theory of compact, oriented Riemannian Einstein 4-manifolds, there is a a fundamental topological constraint that is implied by the Einstein equations. To wit, if $\chi$ and $\...
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Dijkgraaf-Witten topological invariant
We know that given a finite group $G$ and its group 4-cohomology class $w \in H^4[G;U(1)]$, we can obtain a DW topological invariant $Z_{G,w}(M^4)$ as the partition function of the DW theory on a ...
- 4,716
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126
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Relation between characters of a representation and characteristic classes
I have a basic question concerning Atiyah and Schmid's paper "A Geometric Construction of the Discrete Series for Semisimple Lie Groups", Inventiones mathematicae 42 (1977): 1-62".
I will use the ...
- 1,871
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Stiefel-Whitney Classes and Obstructions
Let $E$ be a vector bundle over a simplicial space $B$.
Let $$\mathfrak{o}_k\in H^k(B,\{\pi_{k-1}(V(n,n-k+1))\})$$ be the $k$th obstruction to $k$ independent sections. ($V(n,k)$ the Stiefel manifold....
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Chern Character Number Belongs to integer
From Getzler's definition [1], we know the odd Chern character is the following map
$$Ch:K^1(M)\to H^{odd}(M;\mathbb C),~g\mapsto \sum_{k\geqslant0}(-\frac1{2\pi\...
- 1,905
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Is a complex vector bundle over a punctured closed surface trivial?
Let $M$ be a connected closed surface (possibly with non-zero genus) and let $P\subset M$ be a nonempty finite set of points. Set $\dot{M} = M \setminus P$. Let $\pi : E \rightarrow \dot{M}$ be a ...
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A 4-manifold with Stiefel-Whitney classes $w_1\neq 0$, $w_3 \neq 0$, and $w_1^2=0$?
Is there a 4-manifold whose Stiefel-Whitney classes satisfy $w_1\neq 0$, $w_3 \neq 0$, and $w_1^2=0$?
This question follows on from this one where the condition $w_3 \neq 0$ is replaced by $w_2 \neq ...
- 4,716
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765
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Betti numbers as characteristic numbers?
Let $X$ be a compact differentiable manifold of dimension $m$ or, if you prefer, a smooth complex projective manifold of complex dimension $n=m/2$.
The Euler characteristic $\chi(X):=\Sigma_{i=0}^{m}...
- 22.7k
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2
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Pin$^+$ and Pin$^−$ structure for manifolds in any dimensions
For an oriented $d$-manifold $M$, we can ask whether the manifold
admits a Spin structure, say, if the transition functions for the
tangent bundle, which take values in $SO(d)$, can be lifted to $\...
- 10.3k
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Majorana modes and the first Stiefel–Whitney class
The first Stiefel–Whitney class of a vector bundle is an element in the first cohomology group of the base space. Namely, the first Stiefel–Whitney class for a vector bundle $E$ over a $d$-dimensional ...
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Who discovered this definition of Stiefel-Whitney classes?
I would define Stiefel-Whitney classes as the pullbacks of generators of $H^*(BO, \mathbb{Z}/2)$ under a classifying map, and I gather this is a pretty common definition.
However, the book "...
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4-dimensional TQFT with/without requiring spin structure
My questions here are focused on $D$-dimensional topological quantum field theories (TQFTs) which are unitary and which have finite dimensional Hilbert space on a closed spatial manifold $M^{d-1}$. ...
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Second Stiefel-Whitney class is a square
I'm interested in examples of manifolds which are orientable and such that the second Stiefel-Whitney class is a square. (Of course the second Stiefel-Whitney class should be non-zero.)
An easy ...
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How to calculate the top Chern class of a "functorial" vector bundle on a moduli space of sheaves?
Let $\mathcal M$ be a moduli space of sheaves on a nice variety $X$, with fixed rank $r$ and Chern classes $c_i$, semi-stable with respect to an ample bundle $H$. Let $E$ be a "functorial" vector ...
- 1,990
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Vanishing of characteristic numbers vs vanishing of characteristic classes
A famous result by Thom states that Oriented Bordism classes are determined by characteristic numbers; specifically, two closed manifolds are orientedly bordant if and only if they have the same ...
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Cohomology classes functorial under etale morphisms
Consider the full subcategory $\mathcal C$ in the big etale site of a field $k$ consisting of smooth schemes, namely the category of smooth varieties over a field $k$ with etale maps as morphisms.
I ...
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Second Stiefel-Whitney class as an obstruction to the existence of spin structure
Let $M$ be an oriented (closed) Riemannian manifold. Choose a good open cover and local trivialisations of the tangent bundle $U_i$. Then we get a system of transition functions $\varphi_{ij}: U_i \...
- 9,150
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1
answer
199
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Segre Classes of reducible variety
Suppose I have a singular projective variety $X\subset \mathbb{P}^n$ that is reducible with $X=\bigcup_i X_i$ smooth irreducible components. That is, the irreducible components are smooth but $X$ is ...
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Vector bundle over an oriented manifold with non-vanishing w_2w_3
I am looking for an example of an oriented rank 5 (or lower) real vector bundle $V$ over an oriented manifold such that the cup product $w_2(V) w_3(V)$ of Stiefel-Whitney classes does not vanish. It ...
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What is the first Pontryagin class of the $n$-dimensional representation of $S_n$?
The symmetric group $S_n$ has an $n$-dimensional defining representation, which splits as $n = (n-1) + 1$. Although this representation exists integrally, I would like to think of this as a real ...
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Chern class on a symplectic manifold
Let $(X,\omega)$ be a closed symplectic manifold. Can we always write $c_1(TX) = [ f \omega ]$
for some function $f: X \to \mathbb{R}$?
- 191
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539
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Algebraic relationships between the Stiefel-Whitney classes of a manifold?
I've noticed that in a 2D manifold, the second Stiefel-Whitney class can always be obtained as the cup product of the first one with itself.
In other words $w_2=w_1\smile w_1$.
Is there a 'natural' ...
- 151
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Chern classes via degeneracy loci
According to book of Eisenbud-Harris Page 332 and the following summary http://pbelmans.ncag.info/blog/2014/10/09/what-are-chern-classes/
one can describe Chern classes in terms of degeneracy loci.
...
2
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1
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915
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Infinite Grassmannian does not have the homotopy type of a finite-dimensional complex
Is there a proof that $BO(k)$ is not of the homotopy type of a finite dimensional complex?
The Grassmannian $BO(k) := \{ k\text{-dim subspaces of } \mathbb{R}^\infty \}$ classifies the $k$-...
- 1,062
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350
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Existence of flat connections via characteristic classes, for nice groups
I have two questions about what I write below (which honestly seems pretty elementary).
Is it true (more or less)?
Is there a clean reference that I can cite.
Let $G$ be a compact Lie group, $M$ a ...
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1
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792
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Measuring the failure of pushforward to commute with Steenrod squares
Let $f \colon X \rightarrow Y$ be a map of topological spaces. Let's say that they are (closed) manifolds (not necessarily orientable), though to be honest I'm really interested in the more general ...
- 2,769
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Vector bundles with exactly one nonzero SW-class
I am interested in seeing examples of a space $X$ (preferably a closed smooth manifold, but any finite-dimensional CW-complex would also be of interest) with a vector bundle $\xi\colon E \to X$ on it, ...
- 11.9k
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869
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Local proof of Grothendieck-Riemann-Roch theorem
There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem.
Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of ...
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3
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1k
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Nice things that can be proved easily with characteristic classes
A bit of context for this question: as a project for my master's degree my supervisor asked me to understand the construction of Milnor's exotic spheres. After learning the heavy material (I knew very ...
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Is there a matrix representation of the permutation group whose character is the Markov trace?
Let $g$ be an element in the permutation group (symmetric group) $S_N$. Define the Markov trace of $g$ (denoted $\text{tr}_k g$) as
$$\text{tr}_kg = k^\text{number of cycles in $g$} ,$$
which depends ...
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