A FURTHER GENERALIZATION OF THE SEGAL CONJECTURE

Abstract

THE Segal conjecture has been subjected to a number of usefulgeneralizations over the years [1,3,9,14,19]. We here give a still furthergeneralization which includes those in all of the cited papers. Weconsider an extension1->AT->G->F->1,where G is a compact Lie group, K is a (closed) normal subgroup, andthe quotient G/K = F is finite. There is a classifying F-space B(K; G) inthis situation. We shall prove that the generalized version of the Segalconjecture proven for stable F-cohomotopy in [1] remains valid for stableF-cohomotopy with coefficients in B(K; G). The case G = F x K, Kfinite, was studied in [9], where the close connection between the Segalconjecture and equivariant classifying spaces was first observed. Thispaper is the pushout over [9] of the unpublished preprints [14] and [19],which deal with the cases G finite and G = F x K, respectively.We introduce ideas in Section 1 by explaining the implications of ourresults for the calculation of equivariant stable maps between equivariantclassifying spaces. We state our main theorems in Section 2. We reducethe proofs of the theorems to questions about p -groups F and p-adiccompletion in Section 3. We prove the theorems when G is finite inSection 4. We also observe there that Carlsson's theorem [3] that theSegal conjecture for elementary Abelian p-groups implies the Segalconjecture for all finite p-groups is intrinsically a statement aboutequivariant classifying spaces. We prove our theorems when G is anextension of a torus by a finite group in Section 6. The proof proceeds byreduction to the case when G is finite. It depends on the dualization of acalculation of McClure [16], which may be of independent interest and isgiven in Section 5. We prove the general case of the theorems in Section7. The proof proceeds by reduction to the case handled in Section 6. Weuse results of Feshbach [7] to generalize some of our calculations inSection 8.