#### Volume 16, issue 2 (2012)

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On the Taylor tower of relative $K$–theory

### Ayelet Lindenstrauss and Randy McCarthy

Geometry & Topology 16 (2012) 685–750
##### 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 $\stackrel{̃}{K}\left(R;\stackrel{̃}{M}\left[X\right]\right)$ as a functor of $X$. Resolving general $R$–bimodules by bimodules of the form $\stackrel{̃}{M}\left[X\right]$, this also determines the Goodwillie Taylor tower of $\stackrel{̃}{K}\left(R;M\right)$ as a functor of $M$. The towers converge when $X$ or $M$ is connected. This also gives the Goodwillie Taylor tower of $\stackrel{̃}{K}\left(R⋉M\right)\simeq \stackrel{̃}{K}\left(R;B.M\right)$ 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 $X↦{W}_{n}\left(R;\stackrel{̃}{M}\left[X\right]\right)$ is the $n$–th stage of the Goodwillie calculus Taylor tower of the functor which sends $X↦\stackrel{̃}{K}\left(R;\stackrel{̃}{M}\left[X\right]\right)$. Thus the functor $X↦W\left(R;\stackrel{̃}{M}\left[X\right]\right)$ is the full Taylor tower, which converges to $\stackrel{̃}{K}\left(R;\stackrel{̃}{M}\left[X\right]\right)$ for connected X.

##### Keywords
algebraic $K$–theory, $K$–theory of endomorphisms, Goodwillie calculus of functors
##### Mathematical Subject Classification 2000
Primary: 19D55
Secondary: 55P91, 18G60
##### Publication
Received: 1 March 2008
Revised: 27 October 2011
Accepted: 15 December 2011
Published: 25 April 2012
Proposed: Walter Neumann
Seconded: Haynes Miller, Ralph Cohen
##### Authors
 Ayelet Lindenstrauss Department of Mathematics Indiana University 831 E Third St Bloomington IN 47405 USA Randy McCarthy Department of Mathematics University of Illinois, Urbana 1409 W Green Street Urbana IL 61801 USA