Ordinary I?o(g)-graded Cohomology

Let G be a compact Lie group. What is the appropriate generalization of singular cohomology to the category of G-spaces XI The simplest choice is the ordinary cohomology of EG xG X, where EG is the total space of a universal principal G-bundle. This Borel cohomology [1] is readily computable and has many applications, but is clearly inadequate for such basic parts of G-homotopy theory as obstruction theory. Another choice is Bredon cohomology [2] , as generalized from finite to compact Lie groups by several authors. This gives groups HQ(X;M) for n > 0 and for a "coefficient system" M. Here M is a contravariant functor from the homotopy category of orbit spaces G/H and G-maps to the category Ab of Abelian groups. (Subgroups are understood to be closed.) Bredon cohomology is adequate for obstruction theory. For finite G, Triantafillou has used it to algebraicize rational G-homotopy theory [11], and she and two of us have used it to set up the foundations of the theory of localization of Gspaces for general G [8] . When M is constant at an Abelian group A, written M = A, we have (*) H"G(X;A) = H (X/G;A).