Abstract

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 com-putability, 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 ex-plained in [2], a less crude invariance property can sometimes be exploited to obtain

Citations