Volume 16, issue 2 (2012)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 22
Issue 6, 3145–3760
Issue 5, 2511–3144
Issue 4, 1893–2510
Issue 3, 1267–1891
Issue 2, 645–1266
Issue 1, 1–644

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
On the Taylor tower of relative $K$–theory

Ayelet Lindenstrauss and Randy McCarthy

Geometry & Topology 16 (2012) 685–750

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.

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