Monday, 12 September 2011

at.algebraic topology - Equivariant singular cohomology

In 1965 or so, Glen Bredon defined ordinary equivariant cohomology, ordinary meaning
that it satisfies the dimension axiom: For each coefficient system M (contravariant
functor from the orbit category of G to the category of Abelian groups), there is a
unique cohomology theory HG(;M) such that, when restricted to the orbit category,
it spits out the functor M. Just as in the nonequivariant world, it can be defined
using either singular or cellular cochains, the latter defined using G-CW complexes.
This works as stated for any topological group G.



For an abelian group A, Borel cohomology with coefficients in A, H(EGtimesGX;A)
is the extremely special case in which one takes M to be the constant coefficient system
underlineA at the group A and replaces X by EGtimesX. That is,



HG(EGtimesX;underlineA)=H(EGtimesGX;A)

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