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)].

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 |