Questions tagged [characteristic-classes]
Cohomology classes associated to vector bundles. Includes Stiefel-Whitney classes, Chern classes, Pontryagin classes, and the Euler class.
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Vanishing of Euler class
Given a real oriented vector bundle E over the base space B of rank n, such that the Euler characteristic class in the n-th cohomology group of B vanishes, is it true that there exists a global ...
81
votes
3
answers
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Intuitive explanation for the Atiyah-Singer index theorem
My question is related to the question Explanation for the Chern Character to this question about Todd classes, and to this question about the Atiyah-Singer index theorem.
I'm trying to learn the ...
20
votes
3
answers
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Non-stably trivial bundle with trivial characteristic classes
Though it's relatively clear that the characteristic classes do not characterise a vector bundle (and after looking through some books) I could not find an example of a vector bundle which is not ...
14
votes
1
answer
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Characteristic classes for odd $K$-theory
There are different models of odd $K$-theory. In one case,
one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a ...
12
votes
2
answers
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Vector bundle for prescribed Stiefel-Whitney classes
I hope this is not trivial.
Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice)
For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology ...
8
votes
2
answers
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rational cohomology of finite real grassmannian
Let $p_j$ to be the $j$-th Pontryagin class of the universal $n$-plane bundle $E_n(\mathbb{R}^\infty)\to G_n(\mathbb{R}^\infty)$. Then according to Theorem 1.6, The Cohomology of BSO n and BO n with ...
7
votes
2
answers
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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 ...
6
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0
answers
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The Stiefel-Whitney and Pontryagin classes of SO(3)-bundles
Consider $SO(n)$ bundles over smooth manifolds. Then using the fact that the Stiefel-Whitney classes are the modulo 2 reductions of the Chern classes, one can prove $w_{2i}^2(E) = p_i(E) \bmod 2$. Now ...
53
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4
answers
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Explanation for the Chern character
The Chern character is often seen as just being a convenient way to get a ring homomorphism from K-theory to (ordinary) cohomology.
The most usual definition in that case seems to just be to define ...
48
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6
answers
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Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic
Hello,
I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...
42
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6
answers
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What is geometrically the Pontryagin class?
What does the Pontryagin class detects or is an obstruction to? Please avoid any answer using that it's the even Chern class of the complexified bundle or any interpretation that relies on the ...
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2
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A geometric characterization for arithmetic genus
Let $X$ be a smooth projective variety over $\mathbb{C}$. The following information is all equivalent (any of these numbers can be computed by a linear equation from any of the others):
the ...
36
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4
answers
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Construction of the Stiefel-Whitney and Chern Classes
I've seen two constructions of these characteristic classes. The first comes from Milnor and Stasheff's book and involves the Thom isomorphism and (at least for me) the rather mysterious Steenrod ...
25
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0
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Status of the Euler characteristic in characteristic p
In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin
d'être aussi ...
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5
answers
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Stiefel-Whitney Classes over Integers?
An interesting thing happened the other day. I was computing the Stiefel-Whitney numbers for $\mathbb{C}P^2$ connect sum $\mathbb{C}P^2$ to show that it was a boundary of another manifold. Of course, ...
16
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1
answer
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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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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 ...
15
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1
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Is it true that all real projective space $RP^n$ can not be smoothly embedded in $R^{n+1}$ for n >1
So first for n even, $RP^n$ is not orientable, hence can not be embedded in $\mathbb{R}^{n+1}$.
For odd n, $RP^{n}$ is orientable, hence the normal bundle is trivial. Now using stiefel-Whitney ...
14
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1
answer
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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 ...
14
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1
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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, ...
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 ...
12
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1
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Chern numbers via Euler characteristics?
Let $X$ be a space good enough to have a fundamental class, and $E$ a complex vector bundle on $X$. Let $P$ be some polynomial expression, and say I want to evaluate $P(c_i(E)) \cap [X]$.
Is ...
11
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1
answer
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How to flow submanifolds?
Motivation
We want a consistent way of perturbing a submanifold away from itself. For $0$-dimensional submanifolds, this is the same data as a nowhere-vanishing vector field: we may flow the points ...
11
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2
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first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles
Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class
$$
w_1(\xi)=0
$$
if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to $SO(...
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3
answers
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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 "...
10
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1
answer
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Whitney sum formula for Pontryagin classes I
I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand this ...
9
votes
0
answers
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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 ...
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3
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Second Stiefel Whitney class of quotients of odd spheres
I don't know much of algebraic topology so the following question could be very silly. Let $G$ a finite subgroup of $U(n)$ that acts linearly (the action induced by the action of $U(n)$ on $\mathbb{C}^...
8
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0
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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 ...
8
votes
2
answers
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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 ...
8
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Which map realizes the isomorphism $KO_n(X)\otimes \mathbb{Q}\to \bigoplus_{i\in\mathbb{Z}}H_{n-4i}(X;\mathbb{Q})$?
The description of the real $KO$-homology groups $KO_n(X)$ can be given abstractly as maps to the real K-theory spectrum $KO$ smash $X$, or via triples $(M,x,\phi)$ where $M$ is a closed manifold, $\...
7
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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....
7
votes
1
answer
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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 ...
7
votes
2
answers
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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 ...
6
votes
1
answer
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Whitney sum formula for Pontryagin classes II
I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand the 2-...
6
votes
2
answers
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Tangent bundle of smooth closed simply-connected $4$-manifold $w_1 = w_2 = 0$ can be trivialized in complement of point?
Let $M$ be a smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$. Can $TM$ be trivialized in the complement of a point?
5
votes
1
answer
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Pontryagin number for 4-dim surface bundle
In paper arXiv:math/0701247
"Divisibility of the stable Miller-Morita-Mumford classes" by Soren Galatius, Ib Madsen, Ulrike Tillmann, it was shown
that the Pontryagin numbers for a 4-dim surface ...
5
votes
4
answers
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integral or rational cohomology of real grassmannians
I have obtained that the cohomology rings
$$
H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k].
$$
Also
$$
H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
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0
answers
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characteristic classes of a covering space with symmetric group action
Let $S_n$ be the $n$-th symmetric group. Suppose we have a $n!$-sheeted covering space
$$
S_n\to M\to M/S_n
$$
where $M$ is a manifold. Let $\mathbb{K}$ be the real numbers $\mathbb{R}$, complex ...
5
votes
1
answer
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A vector bundle associated to a codimension $1$ submanifold of a symplectic manifold
We consider the standard symplectic structure $\omega=\sum dx_i\wedge dy_i$ on $\mathbb{R}^{2n}$. To every codimension $1$ submanifold $M\subset \mathbb{R}^{2n}$ we associate a vector bundle ...
5
votes
1
answer
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characteristic classes of tangent bundle of 2-nd unordered configuration space
Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space
$$
B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2
$$
where
$$
\Delta=\{(m,m)\mid m\...
5
votes
1
answer
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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
0
answers
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Relations between Stiefel-Whitney classes on mapping torus
In question
Relations between Stiefel-Whitney classes
the relations between Stiefel-Whitney classes on manifold are obtained.
My question is that do we have additional relations between Stiefel-...
3
votes
0
answers
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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 ...
3
votes
0
answers
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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 ...
3
votes
1
answer
630
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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 $$\mathbb{Z}...
2
votes
1
answer
910
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Infinite Grassmannian does not have the homotopy type of a finite-dimensional complex
Is there a proof that $BO(k)$ is not of the homotopy type of a finite dimensional complex?
The Grassmannian $BO(k) := \{ k\text{-dim subspaces of } \mathbb{R}^\infty \}$ classifies the $k$-...
1
vote
0
answers
164
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Calculation about Chern character in a special setting
I'm confused with working out the Chern character in the following special setting.
Let $E$ be a spinor bundle
$$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$
over sphere $S^{2n}$, where $\rho$ ...
1
vote
1
answer
3k
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When is the first chern class of a Kaehler manifold positive/negative?
I know some examples of compact complex manifolds whose first Chern class does not have a definite sign (is neither negative, nor positive nor zero on all complex curves). I am looking for a necessary ...
1
vote
0
answers
515
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Conventions / Normalizations of Yang-Mills Field Theories
Let the spacetime be 4-dimensional.
In the usual Maxwell theory of Abelian gauge fields $A$, where field strength $F=dA$ one considers the Maxwell action written as
$$
S_{Maxwell}\equiv\int
-\frac{...