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Note that this cohomology ring is most likely not trivial, despite. Since this cohomology ring only depends on the choice of group , it is called the group cohomology of and this coincides with most other notions of group cohomology you might know. Therefore, equivariant cohomology is a generalization of the theory of group cohomologies. The fact that points have interesting cohomology has some enjoyable consequences.
Any with a action has a trivial map to which is equivariant. Now lets consider the case of acting freely and properly on. Since , we get a map from the group cohomology of to the CW cohomology of.
However, when acts freely and properly, the quotient map often goes by a different name: principal -bundle. This is because it is a fiber bundle with each fiber a — torsor.
Characters and cohomology of finite groups | SpringerLink
Principal bundles come up in all sorts of places, in particular because they contain all the important information of a vector bundle with structure group. Thus, we can do the following.
Starting with a vector bundle over some space with structure group , we can reduce it to a principal bundle over note that the quotient. We then forget about the bundle map and only think about the total space and its action which is free and proper. The characteristic homomorphism in this case is a map:. Hence, the vector bundle determined a map from the group cohomology of to the topological cohomology of. If we go a step further, and agree upon some canonical generators for the group cohomology, we can just talk about the images of these generators as cohomology classes on ; these are called characteristic classes.
A brief list of some well-known groups and their characteristic classes:. Does anyone know the group for Steifel-Whitney classes?
Characters and cohomology of finite groups
Anyway, so equivariant cohomology is clearly very neat. It would seem like this is a doomed undertaking, because is infinite-dimensional. Yet, as we will see, it is not only possible, but in some ways more natural, to phrase this theory in terms of a kind of deRham theory. Tags: math. This area has been one of intense research during the s, with major breakthroughs that have illuminated the wa Group cohomology; 2.
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Products and change of group; 3. Relations with subgroups and duality; 4.
Spectral sequences; 5. Representations and vector bundles; 6. Bundles over the classifying space for a discrete group; 7. The symmetric group; 8. Characteristic classes are global invariants that 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. The notion of characteristic class arose in in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds. 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 classes are elements of cohomology groups;  one can obtain integers from characteristic classes, called characteristic numbers.
On the regularity conjecture for the cohomology of finite groups
Some important examples of characteristic numbers are Stiefel—Whitney numbers , Chern numbers , Pontryagin numbers , and the Euler characteristic. From the point of view of de Rham cohomology , one can take differential forms representing the characteristic classes,  take a wedge product so that one obtains a top dimensional form, then integrates over the manifold; this is analogous to taking the product in cohomology and pairing with the fundamental class.
Characteristic numbers solve the oriented and unoriented bordism questions : two manifolds are respectively oriented or unoriented cobordant if and only if their characteristic numbers are equal. Characteristic classes are phenomena of cohomology theory in an essential way — they are contravariant constructions, in the way that a section is a kind of function on a space, and to lead to a contradiction from the existence of a section we do need that variance.
In fact cohomology theory grew up after homology and homotopy theory , which are both covariant theories based on mapping into a space; and characteristic class theory in its infancy in the s as part of obstruction theory was one major reason why a 'dual' theory to homology was sought.
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Related Characteristic Classes and the Cohomology of Finite Groups
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