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.

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)