Talk:Chern class

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Can this be made more accessible? A longer lead would help. Maybe more motivation and some applications? Geometry guy 20:38, 9 June 2007 (UTC)

The hairy ball theorem is a very bad example to give not so much because it concerns the real differentiable case (after all, the tangent bundle to the Riemann sphere is indeed a complex bundle) but because the bundle is stably trivial (in particular, its Stiefel-Whitney classes vanish) and stably trivial bundles are precisely the sort of thing Chern classes cannot detect. --Gro-Tsen 03:41, 17 July 2005 (UTC)

"where H is Poincaré-dual to the hyperplane $\mathbb{CP}^{n - 1} \subseteq \mathbb{CP}^n$ ". How can a class H be dual to the hyperplane $\mathbb{CP}^{n - 1}$ ?

The hyperplane H represents a homology class in

H 2 n - 2

, which is Poincare-dual to

H 2

, which is where the cohomology

c 1

lives. That is,

c 1

of the bundle equals

- P D ([h])

, where

P D

stands for Poincare-dual and

h

is the hyperplane. Is your complaint simply that the text does not say "... dual to the class of the hyperplane..." (which is technically correct but pedantic, I think)? Joshuardavis 16:03, 26 September 2005 (UTC)

[edit] Revert anonymous edit

I reverted this anon edit:

The field strength F may be viewed as a matrix of 4-forms (or 2 2-forms), and thus the eigenvalues of F are 4-forms as well. Each Chern class is then a 4k (2x2k)form.

back to the original

The field strength F may be viewed as a matrix of 2-forms, and thus the eigenvalues of F are 2-forms as well. Each Chern class is then a 2k form.

I'm not sure what the anonymous editor was thinking ... perhaps they were counting the number of spinor indecies instead of vector indecies? Or were they counting two real components instead of one complex component? linas 02:26, 18 October 2005 (UTC)

[edit] Grothendieck or Atiyah?

I'm not sure that it is correct to attribute the splitting principal to Grothendieck. I thought it was due to Atiyah, although I could be wrong. Can anyone provide a reference? 151.204.6.171

The basic Grothendieck paper is here: [1]. 1958 - I'd thought it was a little earlier. If I recall, he has a principle about reduction to Borel subgroups; and his theory is axiomatic. It is possible Atiyah has an earlier published version. Charles Matthews 19:13, 2 November 2005 (UTC)

[edit] Implications of Chern Classes

I know that the Chern classes are obstructions to finding non-vanishing vector fields. What I don't know, and hence i think it would be nice to add to this entry, is what the converse implies. I.e. if I know that the first Chern class vanishes, then what can I conclude? for exemple, if the first Stiefel-Whitney class vanishes then the bundle is orientable - but it doesn't imply that the bundle has a nowhere vanishing section (eg.: the tangent space of the 2-sphere. All of its SW classes vanishes, as it does for any n-sphere, but it doesn't have a nowhere zero section).