\(G \)-theory of \(\mathbb F_1\)-algebras I: the equivariant Nishida problem

We develop a version of \(G \)-theory for an \(\mathbb F_1\)-algebra (i.e., the \(K \)-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 \(K \)-theory for finite pointed groups with our \(G \)-theory. We compute the \(G \)-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 \(\oplus _n\pi _{2n}(\mathbb {S}^G)\) with a pre-\(\lambda \)-ring structure, where G is a finite group and \(\mathbb {S}^G\) is the G-fixed point spectrum of the equivariant sphere spectrum.