Geometry & Topology
- Geom. Topol.
- Volume 16, Number 2 (2012), 685-750.
On the Taylor tower of relative $K$–theory
For a discrete ring, a simplicial –bimodule, and a simplicial set, we construct the Goodwillie Taylor tower of the reduced –theory of parametrized endomorphisms as a functor of . Resolving general –bimodules by bimodules of the form , this also determines the Goodwillie Taylor tower of as a functor of . The towers converge when or is connected. This also gives the Goodwillie Taylor tower of as a functor of .
For a functor with smash product and an –bimodule , we construct an invariant which is an analog of with coefficients. We study the structure of this invariant and its finite-stage approximations and conclude that the functor sending is the –th stage of the Goodwillie calculus Taylor tower of the functor which sends . Thus the functor is the full Taylor tower, which converges to for connected X.
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
Permanent link to this document
Digital Object Identifier
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]
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. https://projecteuclid.org/euclid.gt/1513732407