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Home > Art, Film & Photography > Characteristic Classes: Chern Class, Stiefel-Whitney Class, Euler Class, Characteristic Class, Pontryagin Class, Thom Space
Characteristic Classes: Chern Class, Stiefel-Whitney Class, Euler Class, Characteristic Class, Pontryagin Class, Thom Space

Characteristic Classes: Chern Class, Stiefel-Whitney Class, Euler Class, Characteristic Class, Pontryagin Class, Thom Space


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About the Book

Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Chern Class, Stiefel-whitney Class, Euler Class, Characteristic Class, Pontryagin Class, Thom Space, Chern-weil Homomorphism, Todd Class, Splitting Principle, Segre Class. Excerpt: In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X . The cohomology class measures the extent to which the bundle is "twisted" particularly, whether it possesses sections or not. In other words, characteristic classes are global invariants which measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry and algebraic geometry . Definition Let G be a topological group, and for a topological space X, write b G ( X ) for the set of isomorphism classes of principal G -bundles -bundles . This b G is a contravariant functor from Top (the category of topological spaces and continuous functions ) to Set (the category of sets and functions ), sending a map f to the pullback operation f . A characteristic class c of principal G -bundles is then a natural transformation from b G to a cohomology functor H, regarded also as a functor to Set . In other words, a characteristic class associates to any principal G -bundle P X an element c(P) in H (X) such that, if f: Y X is a continuous map, then c ( f P ) = f c ( P ). On the left is the class of the pullback of P to Y; on the right is the image of the class of P under the induced map in cohomology. Characteristic numbers Characteristic classes are elements of cohomology groups; one can obtain integers from characteristic classes, called characteristic numbers . Respectively: Stiefel-Whitney numbers, Chern numbers, Pontryagin numbers, and the Euler charact...


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Product Details
  • ISBN-13: 9781155523712
  • Publisher: Books LLC
  • Publisher Imprint: Books LLC
  • Height: 152 mm
  • Sub Title: Chern Class, Stiefel-Whitney Class, Euler Class, Characteristic Class, Pontryagin Class, Thom Space
  • Width: 229 mm
  • ISBN-10: 1155523717
  • Publisher Date: 04 May 2010
  • Binding: Paperback
  • Spine Width: 3 mm
  • Weight: 91 gr


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Characteristic Classes: Chern Class, Stiefel-Whitney Class, Euler Class, Characteristic Class, Pontryagin Class, Thom Space
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