# Equivariant cohomology

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A cohomology that takes the action of some group into account. More precisely, an equivariant cohomology in the category of -spaces (that is, topological spaces on which the continuous action of a group is defined) and equivariant mappings is a sequence of contravariant functors (taking values in the category of Abelian groups) and natural transformations

with the following properties: a) equivariantly-homotopic mappings of pairs induce identity homomorphisms of the groups ; b) an inclusion of the form

induces an isomorphism

and c) for every pair the following cohomology sequence is exact:

Let be a universal -fibration and let be the space associated with the universal fibre space with fibre (that is, the quotient space under the action of given by ). Then the functors yield an equivariant cohomology theory; here is an arbitrary cohomology theory.

For any fixed group the collection of groups together with all possible homomorphisms induced by inclusions of subgroups of is usually called the system of coefficients for the -theory. In some cases the functors are uniquely defined by their systems of coefficients (for example, when is finite and for ).

#### References

 [1] G.E. Bredon, "Equivariant cohomology theories" , Springer (1967) [2] W.Y. Hsiang, "Cohomology theory of topological transformation groups" , Springer (1975)