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 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 Ln–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