Equivariant iterated loop space theory and permutative G-categories

Bertrand J. Guillou; J. Peter May

doi:10.2140/agt.2017.17.3259

Equivariant iterated loop space theory and permutative G-categories

Bertrand J. Guillou, J. Peter May

Mathematics

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

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_∞-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.

Original language	English
Pages (from-to)	3259-3339
Number of pages	81
Journal	Algebraic and Geometric Topology
Volume	17
Issue number	6
DOIs	https://doi.org/10.2140/agt.2017.17.3259
State	Published - Oct 4 2017

Bibliographical note

Publisher Copyright:
© 2017, Mathematical Sciences Publishers. All rights reserved.

Funding

Acknowledgements Guillou thanks Nat Stapleton for very helpful discussions leading to the rediscovery of the operad PG, which was in fact first defined, but not used, by Shimakawa [43, Remark, page 255]. May thanks Mona Merling for many conversations and questions that helped clarify ideas. We both thank the referee for helpful suggestions. We also thank Anna Marie Bohmann and Angélica Osorno for pointing out a mistake in the original version. That led to a reworking of this paper and to much of the work in the sequels [31; 12; 13]. It also led to the long delay in the publication of this paper, which is entirely due to the authors and not at all to the referee or editors. We thank them for their patience. Guillou was supported by Simons Collaboration Grant 282316.

Funders	Funder number
Simons Collaboration	282316

Keywords

Equivariant algebraic K-theory
Equivariant infinite loop spaces
Permutative categories

ASJC Scopus subject areas

Geometry and Topology

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Equivariant iterated loop space theory and permutative G-categories

Abstract

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Keywords

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