Volume 17, issue 6 (2017)

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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
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 $V\phantom{\rule{0.3em}{0ex}}$–fold loop $G\phantom{\rule{0.3em}{0ex}}$–spaces to several avatars of a recognition principle for infinite loop $G\phantom{\rule{0.3em}{0ex}}$–spaces. We then explain what genuine permutative $G\phantom{\rule{0.3em}{0ex}}$–categories are and, more generally, what ${E}_{\infty }\phantom{\rule{0.3em}{0ex}}$$G\phantom{\rule{0.3em}{0ex}}$–categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt–Priddy–Quillen theorem as a statement about genuine $G\phantom{\rule{0.3em}{0ex}}$–spectra and use it to give a new, categorical proof of the tom Dieck splitting theorem for suspension $G\phantom{\rule{0.3em}{0ex}}$–spectra. Other examples are geared towards equivariant algebraic $K\phantom{\rule{0.3em}{0ex}}$–theory.

Keywords
equivariant infinite loop spaces, permutative categories, equivariant algebraic K-theory
Mathematical Subject Classification 2010
Primary: 55P42, 55P47, 55P48, 55P91
Secondary: 18D10, 18D50