Equivariant calculus of functors and Z/2-analyticity of real K-theory

TL;DR

This paper develops an equivariant Goodwillie calculus framework for functors from finite G-sets to G-spectra, demonstrating that Real algebraic K-theory is Z/2-analytic and its derivative relates to Real MacLane homology.

Contribution

It extends Goodwillie calculus to an equivariant setting, introducing G-linear and analytic functors, and applies this to Real algebraic K-theory showing its Z/2-analyticity.

Findings

Analytic functors with trivial derivatives map highly connected G-maps to G-equivalences.

Real algebraic K-theory is shown to be Z/2-analytic in the equivariant setting.

The derivative of Real K-theory is Z/2-equivalent to Real MacLane homology.

Abstract

We define a theory of Goodwillie calculus for enriched functors from finite pointed simplicial G-sets to symmetric G-spectra, where G is a finite group. We extend a notion of G-linearity suggested by Blumberg to define stably excisive and rho-analytic homotopy functors, as well as a G-differential, in this equivariant context. A main result of the paper is that analytic functors with trivial derivatives send highly connected G-maps to G-equivalences. It is analogous to the classical result of Goodwillie that "functors with zero derivative are locally constant". As main example we show that Hesselholt and Madsen's Real algebraic K-theory of a split square zero extension of Wall antistructures defines an analytic functor in the Z/2-equivariant setting. We further show that the equivariant derivative of this Real K-theory functor is Z/2-equivalent to Real MacLane homology.