On the derivative of the stable homotopy of mapping spaces. (English) Zbl 1065.55007

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 [M. Bökstedt, G. Carlsson, R. Cohen, T. Goodwillie, W. C. Hsiang and I. Madsen, Duke Math. J. 84, 541–563 (1996; Zbl 0867.19003)]. The derivative of \(Q_+X^K\) was first determined by 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 L. Hesselholt [Math. Scand. 70, No. 2, 193–203 (1992; Zbl 0761.55011)] and 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 J. Rognes [Geom. Topol. 6, 853–887 (2002; Zbl 1066.55009)].
Reviewer: Ioan Pop (Iaşi)


55P65 Homotopy functors in algebraic topology
18G55 Nonabelian homotopical algebra (MSC2010)
55P42 Stable homotopy theory, spectra
55P91 Equivariant homotopy theory in algebraic topology