Some Remarks on the Segal and Sullivan Conjectures

Abstract

Introduction. The recurrent theme of this paper is the concept of invariance introduced in [1]. Let G be a finite group and let hG(Q denote the homotopy category of G-CW complexes. Given a set F, of morphisms of our category, a functor h defined on hG(Q is said to be g-invariant if it carries each element of P, to an isomorphism. Maps in hG(Q are determined by fixed point data. Accordingly we obtain interesting sets of morphisms by considering the so-called SC-equivalences. Here 3C is a collection of subgroups of G and a map f: X -+ Y is an SC-equivalence if H:XH yH is a homotopy equivalence for each H e SC. We then say that h is SC-invariant if it inverts all the SC-equivalences. The smaller SC is, the stronger the statement becomes since less and less data is required of a map to ensure that it is inverted by h. The generalization of the Segal conjecture proved in [1] produces the smallest class of subgroups SC such that the functor S'i7r( )' is SC-invariant. S-17r*( )I denotes pro-group valued equivariant stable cohomotopy localized at a multiplicatively closed subset S of the Burnside ring A(G) and completed at an ideal I C A(G). The class SC admits a description given entirely in terms of S and I but does nonetheless depend on the cohomology theory 7-rG. Indeed while equivariant K-theory enjoys a similar (even stronger) property [2], this is not so for an arbitrary A (G)-module valued theory h G. Our first objective will be to produce a collection 3C(S, I) of subgroups such that S'h( )' is 3C(S, I)-invariant for a large class of cohomology theories h*, namely those representable by G-spectra. Families, i.e. collections I' of subgroups that are closed under taking subconjugates, play a privileged role in our context. This is due to the existence of a space El which is i-equivalent to a point and yet is i-isotropic, namely EJK = 0 unless K e i. It follows that there exists a universal