Operadic multiplications in equivariant spectra, norms, and transfers

Andrew J. Blumberg, Michael A. Hill

Research output: Contribution to journalArticlepeer-review

64 Scopus citations

Abstract

We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N operads, equivariant generalizations of E operads. Algebras in equivariant spectra over an N operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill-Hopkins-Ravenel norms, determined by the operad. Analogously, algebras in equivariant spaces over an N operad provide explicit constructions of certain transfers. This characterization yields a conceptual explanation of the structure of equivariant infinite loop spaces.To explain the relationship between norms, transfers, and N operads, we discuss the general features of these operads, linking their properties to families of finite sets with group actions and analyzing their behavior under norms and geometric fixed points. A surprising consequence of our study is that in stark contract to the classical setting, equivariantly the little disks and linear isometries operads for a general incomplete universe U need not determine the same algebras.Our work is motivated by the need to provide a framework to describe the flavors of commutativity seen in recent work of the second author and Hopkins on localization of equivariant commutative ring spectra.

Original languageEnglish (US)
Pages (from-to)658-708
Number of pages51
JournalAdvances in Mathematics
Volume285
DOIs
StatePublished - Nov 5 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2015 Elsevier Inc.

Keywords

  • Equivariant stable homotopy theory
  • Operads

Fingerprint

Dive into the research topics of 'Operadic multiplications in equivariant spectra, norms, and transfers'. Together they form a unique fingerprint.

Cite this