Relative \(K\)-theory and topological cyclic homology
This is a combined review with the article above [\textit{R. McCarthy}, Acta Math. 179, No. 2, 197-222 (1997; Zbl 0913.19001)]. From the introduction of McCarthy's paper: ``In recent years, the study of the algebraic \(K\)-theory space \(K(R)\) of a ring \(R\) has been approached by the introduction of spaces with a more homological flavor. One collection of such spaces is connected to \(K\)-theory by various trace maps and is particularly effective in measuring the relative \(K\)-theory \(K(f)\) associated to a surjective ring homomorphism \(f:R\to S\) whose kernel is nilpotent. \textit{T. G. Goodwillie}'s main theorem [in: Ann. Math., II. Ser. 124, 347-402 (1986; Zbl 0627.18004)]\ shows that the rational homotopy type of \(K(f)\) can be recovered from cyclic homology. The main theorem of this paper shows that, for any prime \(p\), the \(p\)-adic homotopy type of \(K(f)\) can be recovered from topological homology, \(\text{TC}(f)\).'' More precisely, if \(f:R\to S\) is a map of simplicial rings such that \(\pi_0 f\) is surjective with nilpotent kernel then \[ \text{trc}: K(f)^\wedge_p \overset \simeq \leftrightarrow \text{TC}(f)^\wedge_p \] where \(\text{trc}: K\to \text{TC}\) denotes the cyclic trace of \textit{M. Bökstedt, W. C. Hsiang} and \textit{I. Madsen} [``The cyclotomic trace and algebraic \(K\)-theory of spaces'', Invent. Math. 111, No. 3, 465-539 (1993; Zbl 0804.55004)]. The approach here is based on Goodwillie's calculus of functors. In outline: By an argument of Goodwillie it suffices to prove the theorem for maps of the form \[ f:A\times M\to A \] where \(A\) is a ring, \(M\) is an \(A\)-bimodule and \(A\times M\) is the twisted product with multiplication given by \((a,m)(a', m')=(aa', am'+ ma')\). Now define functors from simplicial sets to spectra by \[ F(X):= K(A\times \widetilde M(X))^\wedge_p \quad \text{ and } \quad G(X):= \text{TC}(A\times \widetilde M(X))^\wedge_p \] where \(\widetilde M(X)\) is the reduced free \(M\)-group generated by the simplicial set \(X\). The main result is a consequence of Goodwillie's theorem : The diagram \[ \begin{tikzcd} F(X)\ar[r,"\text{trc}"]\ar[d] & G(X)\ar[d]\\F(\ast)\ar[r,"\text{trc}" '] & G(\ast)\end{tikzcd}\tag{\(\ast\)} \] is homotopy Cartesian if (1) \(F\) and \(G\) are analytic and satisfy the \(p\)-limit condition, and (2) trc induces a homotopy equivalence on derivatives. (1) is checked in the first three sections. The main work lies in varifying (2). This is done in Section 4 where it is shown that the derivatives of \(F\) and \(G\) are both given by topological Hochschild homology, \(\text{THH} (A\times M(X); M)\). This is a consequence of the equivalence of stable \(K\)-theory and THH for simplicial rings [\textit{B. I. Dundas} and \textit{R. McCarthy}, ``Stable \(K\)-theory and topological Hochschild homology'', Ann. Math., II. Ser. 140, No. 3, 685-701 (1994; Zbl 0833.55007)]\ and the equivalence of stable TC-theory and THH after profinite completion [\textit{L. Hesselholdt}, ``Stable topological cyclic homology is topological Hochschild homology'', Astérisque 226, 175-192 (1994; Zbl 0816.19002)]. Goodwillie conjectured \((*)\) not just for maps of simplicial rings but more generally for rings up to homotopy (or Bökstedt's FSP's). This question is settled by Dundas in the second paper under review. The main idea here is to approximate general FSP's by simplicial rings and apply McCarthy's theorem. The approximation resembles the cosimplicial resolution of a space by Eilenberg-MacLane spaces. The main motivation comes from the wish to understand Waldhausen's \(A\)-theory; this is the \(K\)-theory of \(Q(\Omega X)\), the free infinite loop space on the loops of a space \(X\). As a special case one has: \[ \begin{tikzcd} A(X)_p^\wedge\ar[r]\ar[d] & \text{TC}(X)_p^\wedge \ar[d]\\ K(\mathbb Z[\pi_1X])_p^\wedge\ar[r] & \text{TC}(*)_p^\wedge\end{tikzcd} \] is homotopy Cartesian. This has been used by \textit{J. R. Klein} and \textit{J. Rognes} [``The fiber of the linearization map \(A(*)\to K(\mathbb Z)\)'', Topology 36, No. 4, 829-848 (1997; Zbl 0881.55013)]\ to compute the homotopy groups of the fiber of \(A(*)_p^\wedge\to K(\mathbb Z)_p^\wedge\) roughly through dimensions \(p^3\). As a byproduct Dundas proves that stable \(K\)-theory is THH, generalizing his earlier result with McCarthy quoted above to FSP's. This settles a conjecture by Goodwillie. An integral version of the results reviewed here is in preparation by Dundas, Goodwillie and McCarthy.
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