Chapter 1 - Equivariant cohomology of G-CW-complexes and the Borel construction

Summary

By analogy with the notion of a (co)homology theory on some category of topological spaces one can define the notion of an equivariant or G-(co)homology theory on G-spaces, where G is a compact Lie group. Depending on the given frame and intended purpose one actually might impose different sets of axioms for such a definition (see, e.g., [Bredon, 1967b], [torn Dieck, 1987], [Lee, 1968], [May, 1982], [Seymour, 1982]), but the minimal request would be the G-homotopy invariance of the (co)homology functor and a suitable Mayer-Vietoris long exact sequence. These two requirements suffice to get an elementary comparison theorem for G-(co)homology theories similar to the usual non-equivariant case; i.e. if : τ h_G → k_G is a natural transformation between G-(co)homology theories, which is an isomorphism on ‘G-points’ (i.e. homogeneous spaces G/K, K a closed subgroup of G) then τ(X) is an isomorphism for all G-spaces X, which can be obtained from (finitely many) ‘G-points’ by a finite number of the following steps (in any order):

1. replacing a G-space by a G-homotopy equivalent G-space;

2. taking finite coproducts (topological sums) of G-spaces;

3. taking homotopy pushouts (double mapping cylinders of G-maps between G-spaces);

(see [Seymour, 1982] for more sophisticated versions of the comparison theorem). The category obtained this way is just the category of G-spaces, which are G-homotopy equivalent to finite G-CW-complexes (cf. [Puppe, D., 1983] for a discussion of this and related questions in the non-equivariant case, not restricted to finite CW-complexes).