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Algebraic K-theory of spaces, a manifold approach. (English) Zbl 0595.57026
Current trends in algebraic topology, Semin. London/Ont. 1981, CMS Conf. Proc. 2, Part 1, 141-184 (1982).
[For the entire collection see Zbl 0538.00016.]
This paper investigates the relation between the K-theory of spaces functor and the stable concordance spaces functor. More specifically, a natural transformation $$A(X)\to Wh^{CAT}(X)$$ is constructed, where A(X) denotes the algebraic K-theory of the space X, $$Wh^{CAT}(X)$$ is the Whitehead space of X, a certain double deloop of the stable concordance space of X, and CAT denotes the geometric category in question: DIFF, resp. PL or TOP. The fibre of this natural transformation is a homology theory. Combining this with other known results one obtains a splitting up to homotopy $A(X)\simeq \Omega^{\infty}S^{\infty}(X_+) \times Wh^{DIFF}(X) \times \mu (X),$ where $$\mu$$ (X) is a certain homology theory which has since been shown to vanish [cf. the author, Algebraic K- theory of spaces, concordance and stable homotopy theory (to appear)].
The paper gives a very explicit description of the natural transformation $$A(X)\to Wh^{CAT}(X)$$ in terms of spaces of manifolds. From the explicitness of this description it is rather easy to deduce that the composite map $BO \to BG \to A(*) \to Wh^{DIFF}(*)$ is trivial. This leads to certain numerical consequences about the image of the J homomorphism in the algebraic K-theory of the integers. Further it follows that there is a map $$G/O\to \Omega Wh^{DIFF}(*)$$. This map is actually a rational homotopy equivalence [cf. M. Bökstedt, Lect. Notes Math. 1051, 25-37 (1984; Zbl 0589.57032)].
Reviewer: W.Vogell

##### MSC:
 57R50 Differential topological aspects of diffeomorphisms 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds 55P42 Stable homotopy theory, spectra 55Q50 $$J$$-morphism