In this chapter we set forth the main general machinery in the cohomology - and rational homotopy - theory of actions of torus and p-torus groups. The principal result is Theorem (3.1.6), the Localization Theorem of Borel, torn Dieck, W.-Y. Hsiang and Quillen: a greater part of the chapter, however, is devoted to developing some of the immediate consequences of this theorem. In keeping with our general policy most of the results of this chapter will be stated and proved initially for finite-dimensional G-CW-complexes so that only singular cohomology theory need be used. For the sake of reference, however, we shall always indicate for what more general G-spaces the results hold when the cohomology theory used is that of Čech or Alexander-Spanier. Since all G-spaces under consideration will be paracompact, the latter is also the cohomology theory associated with the Eilenberg-MacLane spectrum, and it is equivalent to sheaf-theoretic cohomology as defined in [Bredon, 1967a] or [Godement, 1958].

It is important to observe that all the results of this chapter are G-homotopy invariant. Thus, for example, Theorem (3.1.6) holds for G-spaces which are G-homotopy equivalent to finite-dimensional G-CW-complexes (with the given finiteness conditions on the number of orbit types).

At first, general G-spaces are allotted entire sections to themselves, i.e. Sections 3.2 and 3.4. In Sections 3.5–3.8 general G-spaces are discussed in remarks at the end of the section. In Sections 3.9–3.11 general G-spaces and G-CW-complexes are treated simultaneously.