296:-- 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.
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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.
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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
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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
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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.
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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
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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
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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
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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
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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.
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of the cohomology classes. To be consistent with other articles, and to avoid confusion, it may be better to replace them by the symbol "
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The references should help; I’m familiar with Milnor and
Stasheff, which should help. Allen Hatcher’s book is available free online.
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classic Borel-Serre and
Grothendieck papers where this was all done, now available from Numdam as "Le théorème de Riemann-Roch" (
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https://web.archive.org/web/20171124205559/https://www.ma.utexas.edu/users/a.debray/lecture_notes/u17_characteristic_classes.pdf
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In addition, rather than saying "one can change H to some other letter", how about references to K-Theory and the Chow ring?
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Beyond references, I’d suggested contacting a local math department, as talking with someone may prove very helpful.
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Its only definition of characteristic classes is an extremely abstract one. Such a definition should be mentioned
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Yes, the definition is too general as it stands. However, both to make this definition understandable
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Knowledge. If you would like to participate, please visit the project page, where you can join
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should be mentioned and placed in both the obstruction theory and classifying space frameworks.
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Some specific down-to-earth cases would be valuable here (not just at other related articles).
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The article as it stands is an example of the worst possible kind of
Knowledge math article.
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example should be treated lovingly with considerable detail before continuing.
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The casual mention of
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section certainly is no substitute for the suggestions above!
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The first examples of characteristic classes should be
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