Calculus. III: Taylor series.

*(English)*Zbl 1067.55006It is more than twenty years ago that I heard the author lecture on the topic of the paper. Originally he introduced calculus of homotopy functors as a tool to study pseudo-isotopy spaces of manifolds or related constructions such as Waldhausen’s \(K\)-theory of topological spaces (e.g. see [G. E. Carlsson, R. L. Cohen, T. G. Goodwillie and W. C. Hsiang, K-Theory 1, 53–82 (1987; Zbl 0649.55001)] and [ M. Bökstedt, G. E. Carlsson, R. L. Cohen, T. G. Goodwillie, W. C. Hsiang and I. Madsen, Duke Math. J. 84, 541–563 (1996; Zbl 0867.19003)]). It soon turned out that the methods described by the author or variants of those have a wide range of applications. So the program led people to look for different versions of calculus which are suited for dealing with the particular problems they were investigating (e.g. see [B. Johnson and R. McCarthy, J. Pure Appl. Algebra 137, 253–284 (1999; Zbl 0929.18007)] and [M. Weiss, Trans. Am. Math. Soc. 347, 3743–3796 (1995; Zbl 0866.55020 )]). It took a long time before written accounts of calculus of homotopy functors were available: Calculus I [T. G. Goodwillie, K-Theory 4,1–27, (1990; Zbl 0741.57021)] appeared in 1990. It studies relative stable isotopy spaces using the concept of the “linearization” of a homotopy functor, in other words, its derivative. In 1992 Calculus II appeared [T. G. Goodwillie, K-Theory 5, 295–332 (1992; Zbl 0776.55008 )]. Analytic functors were introduced and some basic examples were studied, among them the identity functor on \(\mathcal{U}\), the category of topological spaces, and Waldhausen’s algebraic \(K\)-theory of topological spaces. The paper [T. G. Goodwillie, Proc. Int. Congr. Math., Vol. I, 621–630 (1991; Zbl 0759.55011)] contains the author’s talk at the International Congress of Mathematicians in Kyoto, Japan, 1991. It contains a proof of the equivalence of relative Waldhausen \(K\)-theory and relative topological cyclic homology using calculus of homotopy functors. The present paper is a continuation of Calculus I and Calculus II. Though it appeared very late, much to the inconvenience of many interested mathematicians, the delay of its publication had its merits: the theory has evolved over the last two decades and the paper reflects this development. It is very well written. A detailed introduction gives an overview of the present paper as well as the previous two parts the basic ideas behind the theory and the basic motivations. The reader is not confronted by a finished picture but is made to participate into its drawing. Let \(\mathcal{U}_Y\), \(\mathcal{T}_Y\), and \(\mathcal{S}p\) denote the categories of topological spaces over \(Y\) \linebreak without and with sections, and the category of spectra, respectively. A homotopy functor \(F\) between these categories is a functor preserving weak homotopy equivalences. It is n-excisive if it takes strongly homotopy cocartesian \((n+1)\)-cubical diagrams to homotopy cartesian diagrams. \(F\) is in \(E_n(c,\kappa)\) if for any strongly homotopy cocartesian \((n+1)\)-cubical diagrams \(\mathcal{X}\) the diagram \(F(\mathcal{X})\) is homotopy cartesian up to dimension \((-c+\Sigma k_s)\), where \(k_1,\ldots,k_{n+1}\) denote the connectivities of the maps from the initial vertex of the cube to its neighboring vertices. \(F\) is stably \(n-excisive\) if it is in some \(E_n(c,\kappa)\), and \(\rho\)-analytic, if there is a number \(q\) such that \(F\) is in \(E_n(n\rho -q, \rho + 1)\) for all \(n\geq 1\). In the first chapter the author constructs the Taylor tower
\[
\cdots P_nF\overset{q_nF} {\rightarrow}P_{n-1}F\overset{q_{n-1}F} {\rightarrow} \cdots \overset{q_1F} {\rightarrow}P_0F
\]
of a homotopy functor from \(\mathcal{U}_Y\) or \(\mathcal{T}_Y\) to \(\mathcal{U}_Y\), \(\mathcal{T}_Y\), or \(\mathcal{S}p\) together with natural maps \(p_nF: F\to P_nF\). The functor \(P_nF\) is the universal approximation (in the homotopy sense) of \(F\) by an \(n\)-excisive functor. If \(F\) is \(\rho\)-analytic and the structure map \(X\to Y\) is \((\rho + 1)\)-connected the \(p_nF\) define an equivalence from \(F(X)\) to the homotopy limit of the tower evaluated at \(X\to Y\). A homotopy functor \(F\) is n-reduced if \(P_{n-1}F\sim \ast\) and n-homogeneous if it is \(n\)-excisive and \(n\)-reduced. The layers \(D_nF=\) hofiber\((P_nF\to P_{n-1}F)\) are always \(n\)-homogeneous. Chapters 2 and 3 study homogeneous functors: they all arise from homogeneous functors to \(\mathcal{S}p\) and are classified in terms of symmetric multilinear functors. The classification uses the \(n\)-th cross effect which takes a homotopy functor \(F\) to a symmetric homotopy functor \(cr_nF\) of \(n\) variables. Chapters 5 and 6 treat the differentials \(D^{(n)}F\) of \(F\), which are the symmetric multilinear functors corresponding to the layers \(D_nF\) of the tower of \(F\). They are shown to be equivalent to the multilinearization of \(cr_nF\). Chapter 7, 8, and 9 are worked examples, the functors \(X\to \Sigma^\infty Map(K,X)_+\), where \(K\) is a finite CW complex, the identity functor \(\mathcal{T} \to \mathcal{T}\), and Waldhausen’s algebraic \(K\)-theory of spaces.

Reviewer: Rainer Vogt (Osnabrück)

##### MSC:

55P99 | Homotopy theory |

55U99 | Applied homological algebra and category theory in algebraic topology |

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