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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 K̃(R;M̃[X]) as a functor of X. Resolving general R–bimodules by bimodules of the form M̃[X], this also determines the Goodwillie Taylor tower of K̃(R;M) as a functor of M. The towers converge when X or M is connected. This also gives the Goodwillie Taylor tower of K̃(R M) K̃(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 XWn(R;M̃[X]) is the n–th stage of the Goodwillie calculus Taylor tower of the functor which sends XK̃(R;M̃[X]). Thus the functor XW(R;M̃[X]) is the full Taylor tower, which converges to K̃(R;M̃[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