Abstract
Let G be a finite group. To any family F of subgroups of G, we associate a thick circle times-ideal F-Nil of the category of G -spectra with the property that every G -spectrum in F-Nil (which we call F-nilpotent) can be reconstructed from its underlying H-spectra as H varies over F. A similar result holds for calculating G-equivariant homotopy classes of maps into such spectra via an appropriate ...
Abstract
Let G be a finite group. To any family F of subgroups of G, we associate a thick circle times-ideal F-Nil of the category of G -spectra with the property that every G -spectrum in F-Nil (which we call F-nilpotent) can be reconstructed from its underlying H-spectra as H varies over F. A similar result holds for calculating G-equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition E is an element of F-Nil 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 G-equivariant E-homology and cohomology, and generalizations of Quillen's F-P-isomorphism theorem when E is a homotopy commutative G-ring spectrum. We show that the subcategory F-Nil contains many G-spectra of interest for relatively small families F. These include G-equivariant real and complex K-theory as well as the Borel-equivariant cohomology theories associated to complex-oriented ring spectra, the L-n-local sphere, the classical bordism theories, connective real K-theory and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family for which these results hold.
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