Volume 23, issue 2 (2019)

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Derived induction and restriction theory

Akhil Mathew, Niko Naumann and Justin Noel

Geometry & Topology 23 (2019) 541–636
Abstract

Let $G$ be a finite group. To any family $\mathsc{ℱ}$ of subgroups of $G\phantom{\rule{0.3em}{0ex}}$, we associate a thick $\otimes$–ideal ${\mathsc{ℱ}}^{Nil}$ of the category of $G$–spectra with the property that every $G$–spectrum in ${\mathsc{ℱ}}^{Nil}$ (which we call $\mathsc{ℱ}$–nilpotent) can be reconstructed from its underlying $H$–spectra as $H$ varies over $\mathsc{ℱ}\phantom{\rule{0.3em}{0ex}}$. 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 $E\in {\mathsc{ℱ}}^{Nil}$ 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 ${\mathsc{ℱ}}_{p}$–isomorphism theorem when $E$ is a homotopy commutative $G$–ring spectrum.

We show that the subcategory ${\mathsc{ℱ}}^{Nil}$ contains many $G$–spectra of interest for relatively small families $\mathsc{ℱ}\phantom{\rule{0.3em}{0ex}}$. 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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Keywords
equivariant homotopy theory, Artin's theorem, Brauer's theorem, induction, spectral sequences, K–theory, topological modular forms, tensor triangulated categories, Quillen's F–isomorphism theorem, group cohomology
Mathematical Subject Classification 2010
Primary: 19A22, 20J06, 55N91, 55P42, 55P91
Secondary: 18G40, 19L47, 55N34