Talk:Characteristic class

New To Advanced Math
Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as characteristic classes, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006


 * Hi beno,
 * The references should help; I’m familiar with Milnor and Stasheff, which should help. Allen Hatcher’s book is available free online.
 * Beyond references, I’d suggested contacting a local math department, as talking with someone may prove very helpful.
 * —Nils von Barth (nbarth) (talk) 17:03, 24 July 2009 (UTC)

rapprochement?

 * section added. —Nils von Barth (nbarth) (talk) 17:03, 24 July 2009 (UTC)

is rapprochement a word?

In the Concise Oxford Dictionary.

Charles Matthews 21:52, 6 Feb 2004 (UTC)


 * Yes – see rapprochement.
 * —Nils von Barth (nbarth) (talk) 17:03, 24 July 2009 (UTC)

Definition
The discussion of characteristic classes seems quite well-written. But I am *not* happy with leaving the functorial definition as the only definition -- this is way too abstract, and general, for most people to get any kind of intuitive feel for the concept.

Some specific down-to-earth cases would be valuable here (not just at other related articles).Daqu 12:08, 17 January 2006 (UTC)

Yes, the definition is too general as it stands. However, both to make this definition understandable and discuss other definitions, the article should discuss a few key examples of characteristic classes. My proposals would be Chern classes, Pontryagin classes and Whitney classes (surprising, huh?). Then the abstract definition can be brought down to earth by (i) making explicit the classifying spaces that represent the functors in question and discussing the explicit cohomology rings of them as the sources of the pull-backs to charcateristic classes; (ii) discussing the corresponding characteristic classes of vector bundles and in particular Chern classes from the viewpoint of differential geometry and curvature of connection in the bundle; (iii) discussing the intersection-theoretic viewpoint (in algebraic geometry, say) that takes the cohomology class of a divisor associated to a line bundle as the starting point and constructs the higher classes essentially algebraically. Details on (ii) and (iii) together with Chern characters, Todd classes and all that should go into the article Chern classes. Stca74 12:50, 15 May 2007 (UTC)

-- A less special definition that is probably easier to grasp is the following: A characteristic class assigns to each topological space a ring homomorphism from the K-ring K(X) to the cohomology ring H(X) that is natural in the sense that it commutes with maps from X to Y. More technical: A characteristic class is a natural transformation from the K-functor to the H-functor (Cohomology functor). I feel that this is closer connected to the intuition behind the Chern-Weil-theory: A characteristic class assigns to a vector bundle a closed differential form (or a sum of such), i.e. some local expression which may be integrated over. — Preceding unsigned comment added by 171.65.239.54 (talk) 23:45, 15 October 2013 (UTC)

Riemann Roch, etc.
This article doesn't discuss, or even reference, the relationship of characteristic classes to the Riemann Roch theory. For example, one should include Grothendieck's generalization of Chern's definition of characteristic classes by means of the splitting principle for vector bundles and perhaps the classic Borel-Serre and Grothendieck papers where this was all done, now available from Numdam as "Le théorème de Riemann-Roch" (http://www.numdam.org/numdam-bin/search?h=nc&id=BSMF_1958__86__97_0&format=complete) and "La théorie des classes de Chern" (http://www.numdam.org/numdam-bin/search?h=nc&id=BSMF_1958__86__137_0).

In addition, rather than saying "one can change H to some other letter", how about references to K-Theory and the Chow ring? Rwilsker 13:41, 18 August 2006 (UTC)

cup product
The article says the following:

"Formally, given $$i_1,\dots,i_l$$ such that $$\sum \mbox{deg}\,c_{i_j} = n$$, the corresponding characteristic number is: $$c_{i_1}\cup c_{i_2}\cup \dots \cup c_{i_m}([M])$$"

By the "$$\cup$$" between the characteristic classes I am guessing it was meant the cup product of the cohomology classes. To be consistent with other articles, and to avoid confusion, it may be better to replace them by the symbol "$$\smile$$". - Subh83 (talk &#124; contribs) 17:26, 2 January 2014 (UTC)
 * I agree with you. I have already used the symbol $$\smile$$ in "quantum cohomology" and "cohomology ring."--Enyokoyama (talk) 05:38, 4 January 2014 (UTC)
 * great, thanks. Have fixed it on Pontryagin class article as well. - Subh83 (talk &#124; contribs) 21:05, 4 January 2014 (UTC)

Pathetically bad article
The article as it stands is an example of the worst possible kind of Wikipedia math article.

Its only definition of characteristic classes is an extremely abstract one. Such a definition should be mentioned last after far more concrete ones.

The first examples of characteristic classes should be Stiefel-Whitney classes for a n-dimensional vector bundle over a compact connected simplicial complex, defined as obstructions to finding a section of the appropriate bundle (with fibre equal to a Stiefel variety of k orthonormal vectors in the vector space original fibre) over the (n-k+1)-skeleton of the base space. This example should be treated lovingly with considerable detail before continuing.

Next, it is most helpful to define a characteristic class of a bundle as the pullback, of a cohomology class of the appropriate classifying space, by a classifying map for the bundle. The connection between Stiefel-Whitney classes as defined above, and this definition, should be explained clearly. Then — at minimum — Chern classes and Pontrjagin classes should be mentioned and placed in both the obstruction theory and classifying space frameworks.

Only then would it be appropriate to mention characteristic numbers. It is utterly ridiculous to skip over the work of describing characteristic classes clearly and jump immediately into defining characteristic numbers — without any concrete examples of characteristic classes are even mentioned.

The casual mention of Stiefel-Whitney, Chern, and Pontrjagin classes in the Motivation section certainly is no substitute for the suggestions above!Daqu (talk) 21:35, 26 August 2015 (UTC)

Improving Article
Some of the material in https://web.archive.org/web/20171124205559/https://www.ma.utexas.edu/users/a.debray/lecture_notes/u17_characteristic_classes.pdf should be included in this article. — Preceding unsigned comment added by 50.246.213.170 (talk) 20:57, 24 November 2017 (UTC)