Multiplicative equivariant K-theory and the Barratt-Priddy-Quillen theorem

Bertrand J. Guillou, J. Peter May, Mona Merling, Angélica M. Osorno

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in [13]. The proof uses a multiplicative elaboration of an additive equivariant infinite loop space machine that manufactures orthogonal G-spectra from symmetric monoidal G-categories. The new machine produces highly structured associative ring and module G-spectra from appropriate multiplicative input. It relies on new operadic multicategories that are of considerable independent interest and are defined in a general, not necessarily equivariant or topological, context. Most of our work is focused on constructing and comparing them. We construct a multifunctor from the multicategory of symmetric monoidal G-categories to the multicategory of orthogonal G-spectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of G-spectra in [12].

Original languageEnglish
Article number108865
JournalAdvances in Mathematics
Volume414
DOIs
StatePublished - Feb 1 2023

Bibliographical note

Publisher Copyright:
© 2023 Elsevier Inc.

Keywords

  • K-theory
  • Multicategories
  • Multifunctors
  • Multiplicative equivariant infinite loop spaces
  • Operads

ASJC Scopus subject areas

  • General Mathematics

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