96j:55013 55P91 55N91 55P42 55P60

Greenlees, J. P. C. ^{(}4-SHEF^{)}; May, J. P. ^{(}1-CHI^{)} Equivariant stable homotopy theory. Handbook of algebraic topology, 277–323, North-Holland, Amsterdam, 1995. This is a survey paper on the systematic homotopical study of group actions. The paper is divided into an introduction and eight sections, and contains sixty-five references.

Throughout this paper a group means a compact Lie group G and subgroups H of G are assumed to be closed. After reviewing briefly the concepts of space level G-homotopy theory (Section 1), the authors discuss the basic concepts of spectrum level G-homotopy theory such as G-spectra and G-prespectra indexed on a complete G-universe U (Section 2). (They note that Section 2 is an abridged edition of the equivariant version of a paper by A. D. Elmendorf et al. [in Handbook of algebraic topology, 213–253, North-Holland, Amsterdam, 1995].) In Section 3 the authors introduce the homology and cohomology theories represented by a G-spectrum with virtual real representations of G as gradings, and in particular lay emphasis on the case of fixed point spectra indexed on U^{G}. Section 4 deals with H-fixed point functors and also presents two kinds of the generalized duality theorems. Section 5 extends the ordinary cohomology theory on G-spaces given in Section 1 to cohomology theory on G-spectra with virtual real representations of G as gradings.

The next three sections deal with the localization and completion theorems: Section 6 elaborates how to relate equivariant to nonequivariant homology and cohomology via the universal free G-space and makes two assertions. Section 7 deals with a strategy for proving these assertions. The final section gives a number of important examples of localization and completion theorems.

{For the entire collection see MR 96g:55002}.

Haruo Minami (J-NARA)

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