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

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.