Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum

TL;DR

This paper characterizes the equivariant commutative ring structures of factors in the G-equivariant sphere spectrum's idempotent splitting, revealing their dependence on subgroup lattice and conjugation, with implications for Burnside rings and norm functors.

Contribution

It provides a complete description of the multiplicative structures and norms in the idempotent splitting of the G-sphere spectrum, linking algebraic and homotopical properties.

Findings

Explicit description of equivariant ring structures

Characterization of multiplicative transfers on Burnside ring localizations

Identification of incomplete norm functor sets in the splitting

Abstract

We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. Our results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, our analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any idempotent element, which is of independent interest to group theorists. As an application, we obtain an explicit description of the incomplete sets of norm functors which are present in the idempotent splitting of the equivariant stable homotopy category.