$G_\infty$-ring spectra and Moore spectra for $\beta$-rings

TL;DR

This paper introduces $G_ abla$-ring spectra, a new class of globally equivariant homotopy types with structured multiplication, and explores their relation to Moore spectra and $eta$-rings, revealing new algebraic insights.

Contribution

It defines $G_ abla$-ring spectra and connects their power operations to $eta$-ring structures, offering a novel perspective on equivariant homotopy theory.

Findings

$G_ abla$-ring spectra have structured power operations.

Moore spectra can admit $G_ abla$-ring structures under certain conditions.

A link between equivariant power operations and $eta$-ring structures is established.

Abstract

In this paper, we introduce the notion of $G_\infty$-ring spectra. These are globally equivariant homotopy types with a structured multiplication, giving rise to power operations on their equivariant homotopy and cohomology groups. We illustrate this structure by analysing when a Moore spectrum can be endowed with a $G_\infty$-ring structure. Such $G_\infty$-structures correspond to power operations on the underlying ring, indexed by the Burnside ring. We exhibit a close relation between these globally equivariant power operations and the structure of a $\beta$-ring, thus providing a new perspective on the theory of $\beta$-rings.