On the Taylor tower of relative K–theory

Lindenstrauss, Ayelet; McCarthy, Randy

doi:10.2140/gt.2012.16.685

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On the Taylor tower of relative $K$–theory

Ayelet Lindenstrauss and Randy McCarthy

Geometry & Topology 16 (2012) 685–750

DOI: 10.2140/gt.2012.16.685

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 $\tilde{K} (R; \tilde{M} [X])$ as a functor of $X$ . Resolving general $R$ –bimodules by bimodules of the form $\tilde{M} [X]$ , this also determines the Goodwillie Taylor tower of $\tilde{K} (R; M)$ as a functor of $M$ . The towers converge when $X$ or $M$ is connected. This also gives the Goodwillie Taylor tower of $\tilde{K} (R ⋉ M) ≃ \tilde{K} (R; B . M)$ 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 $W_{n} (F; P)$ and conclude that the functor sending $X \mapsto W_{n} (R; \tilde{M} [X])$ is the $n$ –th stage of the Goodwillie calculus Taylor tower of the functor which sends $X \mapsto \tilde{K} (R; \tilde{M} [X])$ . Thus the functor $X \mapsto W (R; \tilde{M} [X])$ is the full Taylor tower, which converges to $\tilde{K} (R; \tilde{M} [X])$ for connected X.

Keywords

algebraic $K$–theory, $K$–theory of endomorphisms, Goodwillie calculus of functors

Mathematical Subject Classification 2000

Primary: 19D55

Secondary: 55P91, 18G60

References

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

Volume 16, issue 2 (2012)