Geometry & Topology

Derived induction and restriction theory

Akhil Mathew, Niko Naumann, and Justin Noel

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Abstract

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

We show that the subcategory Nil contains many G –spectra of interest for relatively small families . 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
Received: 27 July 2015
Revised: 24 July 2018
Accepted: 29 August 2018
First available in Project Euclid: 17 April 2019

Permanent link to this document
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

Subjects
Primary: 19A22: Frobenius induction, Burnside and representation rings 20J06: Cohomology of groups 55N91: Equivariant homology and cohomology [See also 19L47] 55P42: Stable homotopy theory, spectra 55P91: Equivariant homotopy theory [See also 19L47]
Secondary: 18G40: Spectral sequences, hypercohomology [See also 55Txx] 19L47: Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91] 55N34: Elliptic cohomology

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

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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