$\mathrm{G}$-theory of $\mathbb{F}_1$-algebras I: the equivariant Nishida problem

Abstract

We develop a version of $\mathrm{G}$-theory for an $\mathbb{F}_1$-algebra (i.e., the $\mathrm{K}$-theory of pointed $G$-sets for a pointed monoid $G$) and establish its first properties. We construct a Cartan assembly map to compare the Chu--Morava $\mathrm{K}$-theory for finite pointed groups with our $\mathrm{G}$-theory. We compute the $\mathrm{G}$-theory groups for finite pointed groups in terms of stable homotopy of some classifying spaces. We introduce certain Loday--Whitehead groups over $\mathbb{F}_1$ that admit functorial maps into classical Whitehead groups under some reasonable hypotheses. We initiate a conjectural formalism using combinatorial Grayson operations to address the Equivariant Nishida Problem - it asks whether $\mathbb{S}^G$ admits operations that endow $\oplus_nπ_{2n}(\mathbb{S}^G)$ with a pre-$λ$-ring structure, where $G$ is a finite group and $\mathbb{S}^G$ is the $G$-fixed point spectrum of the equivariant sphere spectrum.