On the algebraic \(K\)-theory of simply connected spaces. (English) Zbl 0867.19003

Bökstedt, M.; Carlsson, G.; Cohen, R.; Goodwillie, T.; Hsiang, W. C.; Madsen, I.

Duke Math. J. 84, No. 3, 541-563 (1996).

This paper gives an interpretation of Waldhausen’s (reduced) functor \(\tilde A(X)\) in terms of more familiar objects of algebraic topology. The main ingredients are the cyclotomic trace invariant of M. Bökstedt, W. C. Hsiang and I. Madsen [Invent. Math. 111, No. 3, 465-539 (1993; Zbl 0804.55004)] and the calculus of functors of T. G. Goodwillie [“The differential calculus of homotopy functors”, in: Proc. Int. Congr. Math., Kyoto 1990, Vol. I, 621-630 (1991; Zbl 0759.55011)].
The main result of the paper is that after \(p\)-completion \(\tilde A(X)\) agrees with \(\widetilde{TC}(X;p)\) for a simply connected space \(X\) of finite type. Moreover, the authors give a description of the \(p\)-completion of \(\widetilde{TC}(X;p)\). This description is then used to describe the \(p\)-completion of \(\widetilde{Wh}(X)\). Applications to the homotopy type of the space of smooth pseudo-isotopies of a manifold and integral versions of the results are discussed.

Reviewer: A.Cap (Wien)

MSC:

19D10	Algebraic \(K\)-theory of spaces
55P42	Stable homotopy theory, spectra
55P65	Homotopy functors in algebraic topology
57R52	Isotopy in differential topology
55N15	Topological \(K\)-theory

Keywords:

cyclotomic trace; calculus of homotopy functors

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