On the Taylor Tower of Relative K-theory

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.

💡 Research Summary

The paper introduces a new invariant W(F;P) associated to a smash‑product functor F and an F‑bimodule P, extending the classical topological restriction homology TR(F) by incorporating coefficient data. After defining W(F;P), the authors construct its finite‑stage approximations Wₙ(F;P), showing that each Wₙ is an n‑excisive functor in the sense of Goodwillie calculus. They prove that the tower {Wₙ(F;P)}ₙ forms a homotopy‑limit tower converging to W(F;P) and that the transition maps are homotopy equivalences.

The central application concerns the case where F is the FSP (functor with smash product) associated to a discrete ring R, and P is the FSP associated to a simplicial R‑bimodule M evaluated on a simplicial set X, denoted M