Derived induction and restriction theory

Abstract

Let $G$ be a finite group. To any family $\mathcal{ℱ}$ of subgroups of $G$, we associate a thick $\otimes$–ideal ${\mathcal{ℱ}}^{Nil}$ of the category of $G$–spectra with the property that every $G$–spectrum in ${\mathcal{ℱ}}^{Nil}$ (which we call $\mathcal{ℱ}$–nilpotent) can be reconstructed from its underlying $H$–spectra as $H$ varies over $\mathcal{ℱ}$. 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 $E\in {\mathcal{ℱ}}^{Nil}$ 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 ${\mathcal{ℱ}}_p$–isomorphism theorem when $E$ is a homotopy commutative $G$–ring spectrum.

We show that the subcategory ${\mathcal{ℱ}}^{Nil}$ contains many $G$–spectra of interest for relatively small families $\mathcal{ℱ}$. 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.