G-Theory of \F_1-Algebras I: the Equivariant Nishida Problem

Details G-Theory of \F_1-Algebras I: the Equivariant Nishida Problem G-Theory of \F_1-Algebras I: the Equivariant Nishida Problem

2011-10-27

arxiv.org/abs/1110.6001v2

Mathematics

Algebraic Topology

24 pages; v2 introduction rewritten, references added

Scientific paper

We develop a version of G-theory for \F_1-algebras 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 also construct combinatorial Grayson operations on them. We discuss how our formalism is relevant to the Equivariant Nishida Problem - it asks whether there are operations on \S^G that endow \oplus_n\pi_{2n}(\S^G) with a pre-\lambda-ring structure, where G is a finite group and \S^G is the G-fixed point spectrum of the equivariant sphere spectrum.

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