On the derivative of the stable homotopy of mapping spaces (Q1770323)

The goal of the paper is to describe an alternative approach to computing the derivative of the functor \(X\to Q_+ X^K\), with \(K\) a finite complex, \(X^K\) the space of unbased maps \(K\to X\) and \(Q_+\) the unreduced stable homotopy functor. When \(K\) is the circle, this functor arises in Waldhausen's algebraic K-theory of spaces [\textit{M. Bökstedt, G. Carlsson, R. Cohen, T. Goodwillie, W. C. Hsiang} and \textit{I. Madsen}, Duke Math. J. 84, 541--563 (1996; Zbl 0867.19003)]. The derivative of \(Q_+X^K\) was first determined by \textit{T. G. Goodwillie} using framed bordism theory in [K-theory 4, No. 1, 1--27 (1990; Zbl 0741.57021)]. Another approach using configuration spaces can be found in papers of \textit{L. Hesselholt} [Math. Scand. 70, No. 2, 193--203 (1992; Zbl 0761.55011)] and \textit{G. Arone} [Trans. Am. Math. Soc. 351, No. 3, 1123--1150 (1999; Zbl 0945.55011)]. Both of these approaches rely on manifold theory (the configuration space approach uses the fact that \(K\) has the homotopy type of a parallelizable manifold with boundary). The approach of the present paper is manifold free. The main result is obtained by the so-called chain rule in the calculus of homotopy functors studied by the author and \textit{J. Rognes} [Geom. Topol. 6, 853--887 (2002; Zbl 1066.55009)].