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.
In @cite_16 Fausk shows that [Thm. A] HKR00 can be generalized in several ways if one makes some additional assumptions. First, Fausk proves the analogue of thm:gen-artin when @math and @math is a compact Lie group. Moreover, Fausk proves thm:gen-artin when @math (or a closely related ring spectrum), @math is a finite group, @math , and @math is torsion-free (e.g., when @math is a good group in the sense of @cite_29 ). Fausk also obtains generalized Brauer induction theorems in these contexts. Fausk's results do not require a finiteness assumption on @math .
@cite_29 @cite_16
1542454678 2591650896
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).