## Geometry & Topology

### Derived induction and restriction theory

#### Abstract

Let $G$ be a finite group. To any family $ℱ$ of subgroups of $G$, we associate a thick $⊗$–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 $E ∈ ℱ Nil$ 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.

#### Article information

Source
Geom. Topol., Volume 23, Number 2 (2019), 541-636.

Dates
Revised: 24 July 2018
Accepted: 29 August 2018
First available in Project Euclid: 17 April 2019

https://projecteuclid.org/euclid.gt/1555466424

Digital Object Identifier
doi:10.2140/gt.2019.23.541

Mathematical Reviews number (MathSciNet)
MR3939042

Zentralblatt MATH identifier
07056050

#### Citation

Mathew, Akhil; Naumann, Niko; Noel, Justin. Derived induction and restriction theory. Geom. Topol. 23 (2019), no. 2, 541--636. doi:10.2140/gt.2019.23.541. https://projecteuclid.org/euclid.gt/1555466424

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