Geometry & Topology

On the Taylor tower of relative $K$–theory

Ayelet Lindenstrauss and Randy McCarthy

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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(RM)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.

Article information

Geom. Topol., Volume 16, Number 2 (2012), 685-750.

Received: 1 March 2008
Revised: 27 October 2011
Accepted: 15 December 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]
Secondary: 55P91: Equivariant homotopy theory [See also 19L47] 18G60: Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]

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


Lindenstrauss, Ayelet; McCarthy, Randy. On the Taylor tower of relative $K$–theory. Geom. Topol. 16 (2012), no. 2, 685--750. doi:10.2140/gt.2012.16.685.

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