$\mathrm{G}$-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 ${\oplus }_n{\pi }_{2n}({\mathbb{S}}^G)$ 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.

27 pages; v2 introduction rewritten, references added; v3 revised according to the referee's comments (to appear in J. Homotopy Relat. Struct.); v4 section numbers and the title changed according to the published (online) version

References in corpus (3)

• Schemes over $\F_1$ and zeta functions
• Picard Groups and Class Groups of Monoid Schemes
• Regulators and cycle maps in higher-dimensional differential algebraic K-theory