Let @math be a finite group. To any family @math of subgroups of @math , we associate a thick @math -ideal @math of the category of @math -spectra with the property that every @math -spectrum in @math (which we call @math -nilpotent) can be reconstructed from its underlying @math -spectra as @math varies over @math . A similar result holds for calculating @math -equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition @math implies strong collapse results for this spectral sequence as well as its dual homotopy colimit spectral sequence. As applications, we obtain Artin and Brauer type induction theorems for @math -equivariant @math -homology and cohomology, and generalizations of Quillen's @math -isomorphism theorem when @math is a homotopy commutative @math -ring spectrum. We show that the subcategory @math contains many @math -spectra of interest for relatively small families @math . These include @math -equivariant real and complex @math -theory as well as the Borel-equivariant cohomology theories associated to complex oriented ring spectra, any @math -local spectrum, the classical bordism theories, connective real @math -theory, and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family such that these results hold.
Let BG be the classifying space of a finite group G. Given a multiplicative cohomology theory E ⁄ , the assignment G 7i! E ⁄ (BG) is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E ⁄ , using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artin's Theorem is proved for all complex oriented E ⁄ : the abelian subgroups of G serve as a detecting family for E ⁄ (BG), modulo torsion dividing the order of G. When E ⁄ is a complete local ring, with residue field of characteristic p and associated formal group of height n, we construct a character ring of class functions that computes 1 E ⁄ (BG). The domain of the characters is Gn,p, the set of n-tuples of elements in G each of which has order a power of p. A formula for induction is also found. The ideas we use are related to the Lubin Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, E ⁄ n- theory, etc. The nth Morava K-theory Euler characteristic for BG is computed to be the number of G-orbits in Gn.p. For various groups G, including all symmetric groups, we prove that K(n)⁄(BG) concentrated in even degrees. Our results about E⁄(BG) extend to theorems about E⁄(EG◊GX), where X is a finite G-CW complex. Let G be a compact Lie group. We present two induction theorems for certain generalized G-equivariant cohomology theories. The theory applies to G-equivariant K-theory K G , and to the Borel cohomology associated with any complex oriented cohomology theory. The coefficient ring of K G is the representation ring R(G) of G. When G is a finite group the induction theorems for K G coincide with the classical Artin and Brauer induction theorems for R(G).