CW decompositions of equivariant CW complexes

Let G be a compact Lie group. A G-cell of dimension n is a space of the form G/H x D, where H is a closed subgroup of G and D is an n-cell. A G-CW complex X (or an equivariant CW complex in the terminology of [9]) is constructed by iterated attaching of G-cells. It is the union of G-spaces X^> such that X^ is a disjoint union of G-cells of dimension 0, that is, orbits G/H, and X' + 1 ) is obtained from X n ) by attaching G-cells of dimension n + 1 along equivariant attaching maps G/H x dD -* X^K The space X^"\ which is called the n-skeleton of X, is thus the union of all G-cells of dimension at most n (the topological dimension of X^ is in general greater than n). For basic facts about G-complexes see the original papers [5] and [3] or the exposition in [9]. For discrete groups G it is well known that every G-CW complex is also a CW complex with a cellular action of G (this follows for example from [9, Proposition 1.16, p. 102]). For non-discrete groups, Illman [4] gave an example showing that a G-CW complex X does not always admit a CW decomposition, compatible with the given GCW decomposition, and proved that there always exists a homotopy equivalent CW complex Y which is finite if X is a finite G-complex. In this paper we consider the following problem. Given a G-CW complex X, does there exist a G-space Y, G-homotopy equivalent to X, with a CW decomposition such that the action p: G xY -> Y is a. cellular map with respect to some decomposition of G. The existence of such a Y is interesting from the point of view of equivariant homology and cohomology. For example, Greenlees and May showed that for some groups G the generalised Tate cohomology defined in [1] can be calculated from the CW decomposition