Equivariant iterated loop space theory and permutative G–categories

Guillou, Bertrand; May, Peter

doi:10.2140/agt.2017.17.3259

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Equivariant iterated loop space theory and permutative $G$–categories

Bertrand J Guillou and J Peter May

Algebraic & Geometric Topology 17 (2017) 3259–3339

DOI: 10.2140/agt.2017.17.3259

Abstract

We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for $V$ –fold loop $G$ –spaces to several avatars of a recognition principle for infinite loop $G$ –spaces. We then explain what genuine permutative $G$ –categories are and, more generally, what $E_{\infty}$ – $G$ –categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt–Priddy–Quillen theorem as a statement about genuine $G$ –spectra and use it to give a new, categorical proof of the tom Dieck splitting theorem for suspension $G$ –spectra. Other examples are geared towards equivariant algebraic $K$ –theory.

Keywords

equivariant infinite loop spaces, permutative categories, equivariant algebraic K-theory

Mathematical Subject Classification 2010

Primary: 55P42, 55P47, 55P48, 55P91

Secondary: 18D10, 18D50

References

Publication

Received: 14 July 2012

Revised: 3 March 2017

Accepted: 24 March 2017

Published: 4 October 2017

Authors

Bertrand J Guillou
Department of Mathematics The University of Kentucky Lexington, KY United States

J Peter May
Department of Mathematics The University of Chicago Chicago, IL United States

Volume 17, issue 6 (2017)