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

Abstract

Abstract. We develop a version of G-theory for an 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 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 S G admits operations that endow ⊕ n π 2n (S G ) with a pre-λ-ring structure, where G is a finite group and S G is the G-fixed point spectrum of the equivariant sphere spectrum.