A survey of equivariant stable homotopy theory

Abstract

EQUIVARIANT stable homotopy theory was invented by G. B. Segal in the early 1970s [45]. He was motivated by his work with Atiyah [9] on equivariant K-theory, generalizing an earlier theorem of Atiyah’s on the K-theory of classifying spaces of finite groups to compact Lie groups, and by his work on configuration space and discrete models for iterated loop spaces. His work also suggested to him the “Segal conjecture” (see 4.3), which asserts that the zero-dimensional stable cohomotopy group of the classifying space of a finite group is isomorphic to the completed Burnside ring of the group. The statement is a non-equivariant one, but the methods involved in the eventual proof of the conjecture require heavy use of the equivariant theory. In addition, J. P. May and his collaborators have pointed out that an equivariant version of spectra is the natural device for making spectrum level versions of space level constructions, such as the quadratic or p-adic construction used in defining Steenrod operations. A first attempt to construct an equivariant stable homotopy theory would define the set of stable G-maps (G is finite) from X to Y to be the direct limit l& [Z”X, x” YJG, where the G-actions on Z”X and Z”_Y are obtained by directly suspending the actions on X and Y, and c-1 1’ denotes G-homotopy classes of G-maps. Unfortunately, the theory one obtains this way does not allow for many of the familiar constructions one associates with stable homotopy theory, such as S-duality and transfer. To obtain these, one must allow the group G to act on the “suspension coordinate’*. Precisely, for any representation Vof the group G, one can form the one point compactification S” of V, which becomes a based G-space. S” is a dim( Q-sphere. The stable G-maps from X to Y are now defined to be the direct limit l$ [S’ A X, Sy A ylG, where the actions on Sy A X and S” A Y are diagonal actions, and the direct limit is taken over a certain ordering on the representations, so that V 5 W= V is a summand of IV, This construction permits S-duality and transfer. Other topologists quickly began to study the properties of this theory, notably tom Deck [21]. It gradually became clear that this theory was the natural way to stabilize G-homotopy theory; in this setting, it is possible to construct transfers and S-duality, and to prove an equivariant version of Thorn’s transversality theorem. More naive forms of stabilization do not permit these constructions. Some comments about Frank Adams’ contributions to this subject are in order. He became interested in Segal’s conjecture at an early stage, and in [l] he pointed out the central role played by the spectrum RP ?,” in the study of the conjecture for G = Z/22.