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 ...
Dima's user avatar
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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 ...
Daniel Moskovich's user avatar
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 ...
ARG's user avatar
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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 ...
Sebastian Goette's user avatar
12 votes
2 answers
645 views

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 ...
Oliver Straser's user avatar
8 votes
2 answers
972 views

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 ...
QSR's user avatar
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7 votes
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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 ...
Samuel Monnier's user avatar
6 votes
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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 ...
Manuel's user avatar
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53 votes
4 answers
13k views

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 ...
Sam Derbyshire's user avatar
48 votes
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 ...
zygund's user avatar
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42 votes
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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 ...
ARG's user avatar
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38 votes
2 answers
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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 ...
Charles Staats's user avatar
36 votes
4 answers
5k views

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 ...
solbap's user avatar
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25 votes
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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 ...
Vivek Shende's user avatar
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23 votes
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, ...
Justin Curry's user avatar
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16 votes
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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 ...
Taisong Jing's user avatar
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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 ...
Xiao-Gang Wen's user avatar
15 votes
1 answer
4k views

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 ...
Xiaolei Wu's user avatar
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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 ...
Theo Johnson-Freyd's user avatar
14 votes
1 answer
349 views

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, ...
Jens Reinhold's user avatar
13 votes
3 answers
2k views

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 ...
Tim Campion's user avatar
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12 votes
1 answer
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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 ...
Vivek Shende's user avatar
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11 votes
1 answer
570 views

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 ...
John Wiltshire-Gordon's user avatar
11 votes
2 answers
954 views

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(...
QSR's user avatar
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10 votes
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 "...
user84144's user avatar
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10 votes
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 ...
Mark Grant's user avatar
9 votes
0 answers
432 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 ...
Matthias Ludewig's user avatar
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3 answers
742 views

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}^...
Italo's user avatar
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8 votes
0 answers
261 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 ...
Borromean's user avatar
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8 votes
2 answers
472 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 ...
aglearner's user avatar
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8 votes
0 answers
273 views

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, $\...
EdoardoFossati's user avatar
7 votes
0 answers
422 views

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....
Rene Schipperus's user avatar
7 votes
1 answer
536 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 ...
truebaran's user avatar
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7 votes
2 answers
936 views

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 ...
AlexE's user avatar
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6 votes
1 answer
419 views

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-...
Mark Grant's user avatar
6 votes
2 answers
237 views

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?
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5 votes
1 answer
274 views

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 ...
Xiao-Gang Wen's user avatar
5 votes
4 answers
3k views

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 ...
Shiquan Ren's user avatar
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5 votes
0 answers
238 views

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 ...
QSR's user avatar
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5 votes
1 answer
373 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 ...
Ali Taghavi's user avatar
5 votes
1 answer
327 views

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\...
QSR's user avatar
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5 votes
1 answer
405 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^...
annie marie cœur's user avatar
4 votes
0 answers
323 views

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-...
Xiao-Gang Wen's user avatar
3 votes
0 answers
158 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 ...
annie marie cœur's user avatar
3 votes
0 answers
311 views

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 ...
annie marie cœur's user avatar
3 votes
1 answer
630 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}...
wonderich's user avatar
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2 votes
1 answer
910 views

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 $k$-...
Jan Steinebrunner's user avatar
1 vote
0 answers
164 views

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 $\rho$ ...
Radeha Longa's user avatar
1 vote
1 answer
3k views

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 ...
user avatar
1 vote
0 answers
515 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{...
wonderich's user avatar
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