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 $\tilde{K}\left(R;\tilde{M}\left[X\right]\right)$ as a functor of $X$. Resolving general $R$–bimodules by bimodules of the form $\tilde{M}\left[X\right]$, this also determines the Goodwillie Taylor tower of $\tilde{K}\left(R;M\right)$ as a functor of $M$. The towers converge when $X$ or $M$ is connected. This also gives the Goodwillie Taylor tower of $\tilde{K}\left(R⋉M\right)\simeq \tilde{K}\left(R;B.M\right)$ as a functor of $M$.

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