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On the algebraic $$K$$-theory of simply connected spaces. (English) Zbl 0867.19003
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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##### References:
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