## 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)

### MSC:

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