EQUIVARIANT CALCULUS OF FUNCTORS AND \$\mathbb {Z}/2\$-ANALYTICITY OF REAL ALGEBRAIC \$K\$-THEORY

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 ρ-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 the 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.