Questions tagged [characteristic-classes]

Characteristic classes are invariants of bundles living in the cohomology of the base. The most common examples of characteristic classes are the Chern, Stiefel–Whitney, and Pontryagin classes.

For a topological group $G$, denote the collection of isomorphism classes of principal $G$-bundles on a topological space $X$ by $\operatorname{Prin}_G(X)$. The functor $X \to \operatorname{Prin}_G(X)$ satisfies the criteria of Brown's representability theorem, so there is a space $BG$, called the classifying space of principal $G$-bundles, such that $\operatorname{Prin}_G(X) \cong [X, BG]$.

There is a principal $G$-bundle $EG \to BG$ called the universal principal $G$-bundle; the identification above corresponds to pulling back this bundle. More precisely, for every principal $G$-bundle $E \to X$, there is a map $f_E : X \to BG$ such that $E$ is isomorphic to $f_E^*EG$ and $f_E$ is unique up to homotopy. The map $f_E$ is called the classifying map of $E$.

For any $c \in H^{\ast}(BG; R)$, one can define a characteristic class $c(E) \in H^*(X; R)$ by $c(E) := f_E^*c$. With this definition, it is immediate that the association $E \to c(E)$ is natural: given $g : Y \to X$ continuous, $c(g^*E) = g^*c(E)$.

For real vector bundles, one can take $G = O(k)$, in which case $BO(k) = \operatorname{Gr}_k(\mathbb{R}^{\infty})$. The universal principal $O(k)$-bundle is the tautological bundle and often denoted $\gamma \to \operatorname{Gr}_k(\mathbb{R}^{\infty})$. We have $H^{\ast}(\operatorname{Gr}_k(\mathbb{R}^{\infty}); \mathbb{Z}_2) \cong \mathbb{Z}_2[w_1, \dots, w_k]$ where $\deg w_i = i$. The characteristic classes associated to the $w_i$ are called Stiefel-Whitney classes.

For complex vector bundles, one can take $G = U(k)$, in which case $BU(k) = \operatorname{Gr}_k(\mathbb{C}^{\infty})$. The universal principal $U(k)$-bundle is the complex tautological bundle and often denoted $\gamma^{\mathbb{C}} \to \operatorname{Gr}_k(\mathbb{C}^{\infty})$. We have $H^{\ast}(\operatorname{Gr}_k(\mathbb{C}^{\infty}); \mathbb{Z}) \cong \mathbb{Z}[c_1, \dots, c_k]$ where $\deg c_i = 2i$. The characteristic classes associated to the $c_i$ are called Chern classes.

The above is a more modern way of thinking of characteristic classes. The classical reference is Milnor and Stasheff's Characteristic Classes.

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How to interpret the Euler class?

Although I (hardly) understand the formal definition of the Euler class, I have very little intuition of it. I understand that the Euler class of $E\to X$ is zero if and only if there is a section, but what does it mean that the Euler class is…
Pierre D
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Elementary proof of the fact that any orientable 3-manifold is parallelizable

A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors $V_1(p),...,V_n(p)$ provide a basis for the tangent…
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Has anyone seen this combinatorial identity involving the Bernoulli and Stirling numbers?

Does anyone know a nice (combinatorial?) proof and/or reference for the following identity? $$\left( \frac{\alpha}{1 - e^{-\alpha}} \right)^{n+1} \equiv \sum_{j=0}^n \frac{(n-j)!}{n!} |s(n+1, n+1-j)| \alpha^j \bmod…
Qiaochu Yuan
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When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on this space is determined by its characteristic…
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How to understand the Todd class?

I am reading the article "K-Theory and Elliptic Operators"(http://arxiv.org/abs/math/0504555), which is about Atiyah-Singer index theorem. In page 14 the article discussed the Thom isomorphism: $$\psi:H^{k}(X)\rightarrow H^{n+k}_{c}(E)$$ and…
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Different ways of representing a second cohomology class

There are probably many ways of talking about a second (integral) cohomology class of a smooth, closed, orientable manifold $M$ of dimension $n$. Here are a few, with $\alpha\in H^2(M,\mathbb{Z})$: Using Poincare duality, obtain a homology class…
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Cohomology ring of Grassmannians

I'm reading a paper called An Additive Basis for the Cohomology of Real Grassmannians, which begins by making the following claim (paraphrasing): Let $w=1+w_1+ \ldots + w_m$ be the total Stiefel-Whitney class of the canonical $m$-plane bundle over…
matt
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Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Suppose $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ $c_2(V\otimes L)=c_2(V)+(r-1)c_1(V).c_1(L)+{r \choose 2}c_1(L)^2…
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Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the above is true? Note. More precisely, from Wikipedia:…
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Relation between Stiefel-Whitney class and Chern class

A complex vector bundle of rank n can be viewed as a real vector bundle of rank 2n. From nLab, we have that the second Stiefel-Whitney class of the real vector bundle is given by the first Chern class of the complex vector bundle mod 2: $w_2=c_1$…
Xiao-Gang Wen
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Where do Chern classes live? $c_1(L)\in \textrm{?}$

If $X$ is a complex manifold, one can define the first Chern class of $L\in \textrm{Pic}\,X$ to be its image in $H^2(X,\textbf Z)$, by using the exponential sequence. So one can write something like $c_1(L)\in H^2(X,\textbf Z)$. But if $X$ is a…
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The "Wu formula" and Steenrod algebras

The Wikipedia page on Stiefel-Whitney classes includes the following paragraph: Over the Steenrod algebra, the Stiefel–Whitney classes of a smooth manifold (defined as the Stiefel–Whitney classes of its tangent bundle) are generated by those of…
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If the top Stiefel-Whitney class of a compact manifold is nonzero, must there be another non-vanishing Stiefel-Whitney class?

I was trying to collect some examples of Stiefel-Whitney class computations, just to make myself more familiar with them. It seems from my (relatively short list) that if the top Stiefel-Whitney class is nonzero, there must be another nonzero…
Jason DeVito
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Second Stiefel-Whitney Class of a 3 Manifold

This is exercise 12.4 in Characteristic Classes by Milnor and Stasheff. The essential content of the exercise is to show that $w_2(TM)=0$, where $M$ is a closed, oriented 3-manifold, $TM$ its tangent bundle, and $w_2$ the second Stiefel-Whitney…
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Why $\mathbb{RP}^2$ can not be embedded to $\mathbb{R}^3$?

Is there any answer of this question around basic theory of differentiable manifolds?
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