Characteristic Classes in Borel Cohomology

Let G be a topological group and let EC be a free contractible G-space. The Bore1 construction on a G-space X is the orbit space Xo = EC x G X. When asked what equivariant cohomology is, most people would answer Bore1 cohomology, namely H:(X) = H*(X,). This theory has the claim of priority and the merit of ready computability, and many very beautiful results have been proven with it. However, it suffers from the defects of its virtues. Precisely, it is ‘invariant’, in the sense that a G-map f: X + Y which is a nonequivariant homotopy equivalence induces an isomorphism on Bore1 cohomology. A quick way to see this is to observe that 1 x f: EC xX+ EC x Y is a map of principal G-bundles with base map lxof=fo:Xo-‘Y,, so that fc is a weak homotopy equivalence. As explained in [l], this invariance property is the crudest of a hierarchy of such properties that a theory might have. We shall show how to compute all characteristic classes in any invariant equivariant cohomology theory, the conclusion being that no such theory is powerful enough to support a very useful theory of characteristic classes. As explained in [2], a less crude invariance property can sometimes be exploited to obtain a calculation of equivariant characteristic classes in more powerful theories, such as equivariant K-theory. To establish context, consider a closed normal subgroup Z7 of a topological group r with quotient group G. If Y is a n-free r-space, we think of the orbit projection q : Y-* Y/Z7 as a particular kind of equivariant bundle. It is a principal U-bundle in the usual sense, its base space is a G-space and thus a Z7-trivial r-space, and its projection is a r-map. The classical case is r= G x Z7. Here q is called a principal (G, Ill)-bundle, and there is a theory of associated (G, I;T)-bundles exactly as in the nonequivariant case. For example, if G is a compact Lie group acting smoothly on a differentiable manifold M”, then the tangent n-plane bundle of M is a (G, O(n))bundle and is determined by its associated principal (G, O(n))-bundle. See e.g. [4; 5; 7, V $11 for background. There is a universal bundle E(Z7;r) -+ B(I7;I’) = E(Zl7; r)/Ii’ of the sort just specified. Its total space E(L7;T) is a Z7-free T-CW complex such that the /l-fixed point space E(Z7;r)” is contractible for any closed subgroup A of r such that