# 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.

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### Inclusion of Finite Grassmannian in Infinite Grassmannian

I am doing Milnor-Stasheff's exercise 6-B. which is as follows: Show that the restriction homomorphism $i ∗ : H^p (G_n(R ^∞)) → H^p (G_n(R^ {n+k}))$ is an isomorphism for $p<k$. I was thinking of ...
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### Vanishing Chern classes on vector bundle of $S^2$

Suppose $E\to S^2$ be a complex vector bundle. If $c_1(E)=0$, does it imply that $E$ is a trivial bundle? And why if so? This question is motivated from Audin, Damian: Morse Theory and Floer Homology
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### Milnor's definition of the Stiefel Whitney number

In Milnor's characteristic class, he defines the Stiefel-Whitney class as follows: Let $M$ be a closed possibly disconnected smoooth $n$-manifold. Using mod $2$ coefficients there is a unique ...
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### Is there a relationship between the multiplicity of an index and the “algebraic” multiplicity of a zero of a section from a (complex) vector bundle?

I'm a physicist who is trying to make sense of the relationship between the number of zeros of a section from an associated vector bundle and the Euler characteristic. My interest lies in applications ...
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### Equation relating Chern-Simons and Bott-Chern secondary characteristic forms

Let $\overline{\mathcal E}: 0\to \overline S\to \overline E\to \overline Q\to 0$ be a short exact sequence of Hermitian vector bundles over a complex manifold $X$. Here $\overline E=(E,h^E)$ is a ...
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### Equivariant Chern classes and local coefficients

I am trying to understand the basics of equivariant cohomology in view of applications to the field of crystalline topological insulators. At stake in that field is the very explicit situation of the ...
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### A question about the complex conjugate bundle

If a complex vector bundle is constructed by the complexification of a real vector bundle, say $E=F\otimes \mathbb{C}$, then there's a conclusion that $E$ is isomorphic to its conjugate bundle by ...
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