Let G be a finite group. To any family F of subgroups of G,
we associate a thick β-ideal FNil of the
category of G-spectra with the property that every G-spectrum in
FNil (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 EβFNil 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 Fpβ-isomorphism theorem when E is a
homotopy commutative G-ring spectrum.
We show that the subcategory FNil 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, any Lnβ-local spectrum, 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 such that these results
hold.Comment: 63 pages. Many edits and some simplifications. Final version, to
appear in Geometry and Topolog