On the Taylor tower of relative $K$-theory

Abstract

For R a discrete ring, M a simplicial R–bimodule, and X a simplicial set, we construct the Goodwillie Taylor tower of the reduced K–theory of parametrized endomorphisms K~(R;M~[X]) as a functor of X. Resolving general R–bimodules by bimodules of the form M~[X], this also determines the Goodwillie Taylor tower of K~(R;M) as a functor of M. The towers converge when X or M is connected. This also gives the Goodwillie Taylor tower of K~(R⋉M)≃K~(R;B.M) as a functor of M.

For a functor with smash product F and an F–bimodule P, we construct an invariant W(F;P) which is an analog of TR(F) with coefficients. We study the structure of this invariant and its finite-stage approximations Wn(F;P) and conclude that the functor sending X↦Wn(R;M~[X]) is the n–th stage of the Goodwillie calculus Taylor tower of the functor which sends X↦K~(R;M~[X]). Thus the functor X↦W(R;M~[X]) is the full Taylor tower, which converges to K~(R;M~[X]) for connected X.