On the Taylor Tower of Relative K-theory

TL;DR

This paper constructs an invariant W(F;P) related to relative K-theory, analyzes its structure, and demonstrates that it forms the Taylor tower of a functor approximating reduced K-theory with coefficients, establishing convergence and equivalence via Goodwillie calculus.

Contribution

It introduces the invariant W(F;P), connects it to the Taylor tower of K-theory functors, and proves their equivalence using Goodwillie calculus techniques.

Findings

W(F;P) generalizes TR(F) with coefficients

The functor W(R;M[X]) forms the Taylor tower of reduced K-theory

The Taylor tower converges for connected X and is equivalent to relative K-theory

Abstract

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 for F the FSP associated to a ring R and P the FSP associated to the simplicial R-bimodule M[X] (with M a simplicial R-bimodule, X a simplicial set), the functor sending X to W_n(R;M[X]) is the nth stage of the Goodwillie calculus Taylor tower of the functor which sends X to the reduced K-theory spectrum of R with coefficients in M[X]. Thus the functor sending X to W(R;M[X]) is the full Taylor tower, which converges to the reduced K-theory of R with coefficients in M[X] for connected X. We show the equivalence between relative K-theory of R with coefficients in M[-] and W(R;M[-]) using Goodwillie calculus: we construct a natural transformation between the two functors, both of which are 0-analytic, and show that this natural transformation induces an equivalence on the derivatives at any connected X.