Change of universe functors in equivariant stable homotopy theory

One striking difference between nonequivariant and equivariant stable homotopy is that, in the equivariant context, one must specify those representations with respect to which spectra are to be stable. One may specify stability with respect only to trivial representations (thereby obtaining what is often called the naive equivariant stable category), with respect to all representations (thereby obtaining the full equivariant stable category), or with respect to any intermediate collection of representations closed under direct sums. The chosen family of representations is usually described by specifying an indexing universe. Change of universe functors transform spectra stable with respect to one set of representations into spectra stable with respect to a second set of representations. This is done either by restriction (that is, by forgetting the stability with respect to some representations) or by induction (that is, by altering the spectra so that they become stable with respect to a larger class of representations). The impact of these transformations on the equivariant homotopy groups of spectra should be viewed as an equivariant generalization of the passage between unstable and stable homotopy groups in the nonequivariant context. Three results concerning this impact are given. One describes when change of universe functors are isomorphisms of categories. The second completely describes the impact of an arbitrary induction functor on the first nonvanishing homotopy groups of a bounded-below spectrum. The third gives a spectral sequence which describes the behavior of an arbitrary induction functor on all the homotopy groups of an arbitrary spectrum. Introduction. This paper continues the study begun in [13] and [12] of the equivariant Hurewicz and suspension maps. In [12], it was shown that direct equivariant generalizations of the Freudenthal suspension theorem necessarily suffer from at least one of two defects—either their hypotheses are unduly restrictive, or they describe the effect of suspension only on the bottom nonvanishing homotopy groups rather than on the homotopy groups in a range of dimensions. One of the purposes of the present paper is to introduce a spectral sequence, promised in [12], which ameliorates this 1991 Mathematics Subject Classification: Primary 55M35, 55P42, 55P91, 55T99, 57S15; Secondary 55N91, 55P20, 55Q10, 55Q91.