Derived induction and restriction theory

Mathew, Akhil; Naumann, Niko; Noel, Justin

doi:10.2140/gt.2019.23.541

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Derived induction and restriction theory

Akhil Mathew, Niko Naumann and Justin Noel

Geometry & Topology 23 (2019) 541–636

DOI: 10.2140/gt.2019.23.541

Abstract

Let $G$ be a finite group. To any family $ℱ$ of subgroups of $G$ , we associate a thick $\otimes$ –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 $E \in ℱ^{Nil}$ 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.

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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

References

Publication

Received: 27 July 2015

Revised: 24 July 2018

Accepted: 29 August 2018

Published: 8 April 2019

Proposed: Jesper Grodal

Seconded: Ulrike Tillmann, Haynes R Miller

Authors

Akhil Mathew
Department of Mathematics University of Chicago Chicago, IL United States
http://math.uchicago.edu/~amathew/
Niko Naumann
Fakultät für Mathematik Universität Regensburg Regensburg Germany
http://homepages.uni-regensburg.de/~nan25776/
Justin Noel
Fakultät für Mathematik Universität Regensburg Regensburg Germany
http://nullplug.org

Volume 23, issue 2 (2019)