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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

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 EG–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.

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equivariant infinite loop spaces, permutative categories, equivariant algebraic K-theory
Mathematical Subject Classification 2010
Primary: 55P42, 55P47, 55P48, 55P91
Secondary: 18D10, 18D50
Received: 14 July 2012
Revised: 3 March 2017
Accepted: 24 March 2017
Published: 4 October 2017
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