Algebraic K-theory of spaces.

*(English)*Zbl 0579.18006
Algebraic and geometric topology, Proc. Conf., New Brunswick/USA 1983, Lect. Notes Math. 1126, 318-419 (1985).

[For the entire collection see Zbl 0553.00007.]

1978 the author introduces the algebraic K-theory A(X) of a topological space X which is linked to the algebraic K-theory of the simplicial ring \({\mathbb{Z}}[G(X)]\), where G(X) is the Kan loop group of X, and which can be used to obtain information about the homotopy groups of the stable PL- and Diff concordance spaces.

There are many ways to define A(X) and each definition has its own merits. The central part of the present paper is to prove the equivalence up to homotopy of the various definitions. A(X) is most conveniently described as the algebraic K-theory of a certain category with cofibrations and weak equivalences.

The paper is devided into three major sections: ”Abstract K-theory”, ”The functor A(X)”, and ”The Whitehead space \(Wh^{PL}(X)\), and its relations to A(X)”. The first section contains the fundamentals of the K-theory of categories with cofibrations and weak equivalences. Given such a category \({\mathcal C}\), the author associates with it a simplicial category \(S_*{\mathcal C}\) with cofibrations and weak equivalences and defines the K-theory K\({\mathcal C}=\Omega | w S_*{\mathcal C}|\), where \(| w S_*{\mathcal C}|\) is the topological realization of the simplicial category w \(S_*{\mathcal C}\) of weak equivalences in \(S_*{\mathcal C}\). Categories with cofibrations and weak equivalences and the construction \(S_*\) generalize exact categories and the Q-construction in Quillen’s definition of the K-theory of exact categories, and the author proves that basic properties of the Q-construction such as additivity, reduction by resolutions, and devissage have their counterpart in his setting.

Moreover, results about the interplay between weak and strong notions of weak equivalences provide an additional powerful tool for the proofs in the remaining sections. It seems to me that the very general setting of this sections admits applications in areas of algebraic topology and homological algebra beyond the algebraic K-theory of spaces.

The machinery developed in Section 1 is used in Section 2 to prove the equivalence of various definitions of A(X) and to define the linearization transformation A(X)\(\to K({\mathbb{Z}}[GX]).\)

Section 3 is devoted to an abstract version (in the spirit of Section 1) of the relation of A(X) to concordance theory, represented by the Whitehead space \(Wh^{PL}(X)\). The author defines \(Wh^{PL}(X)\) as the realization of a simplicial category and establishes the homotopy fibration \[ h(X,A(*))\to A(X)\to Wh^{PL}(X) \] where h(X,A(*)) is the homology theory associated to A(*).

1978 the author introduces the algebraic K-theory A(X) of a topological space X which is linked to the algebraic K-theory of the simplicial ring \({\mathbb{Z}}[G(X)]\), where G(X) is the Kan loop group of X, and which can be used to obtain information about the homotopy groups of the stable PL- and Diff concordance spaces.

There are many ways to define A(X) and each definition has its own merits. The central part of the present paper is to prove the equivalence up to homotopy of the various definitions. A(X) is most conveniently described as the algebraic K-theory of a certain category with cofibrations and weak equivalences.

The paper is devided into three major sections: ”Abstract K-theory”, ”The functor A(X)”, and ”The Whitehead space \(Wh^{PL}(X)\), and its relations to A(X)”. The first section contains the fundamentals of the K-theory of categories with cofibrations and weak equivalences. Given such a category \({\mathcal C}\), the author associates with it a simplicial category \(S_*{\mathcal C}\) with cofibrations and weak equivalences and defines the K-theory K\({\mathcal C}=\Omega | w S_*{\mathcal C}|\), where \(| w S_*{\mathcal C}|\) is the topological realization of the simplicial category w \(S_*{\mathcal C}\) of weak equivalences in \(S_*{\mathcal C}\). Categories with cofibrations and weak equivalences and the construction \(S_*\) generalize exact categories and the Q-construction in Quillen’s definition of the K-theory of exact categories, and the author proves that basic properties of the Q-construction such as additivity, reduction by resolutions, and devissage have their counterpart in his setting.

Moreover, results about the interplay between weak and strong notions of weak equivalences provide an additional powerful tool for the proofs in the remaining sections. It seems to me that the very general setting of this sections admits applications in areas of algebraic topology and homological algebra beyond the algebraic K-theory of spaces.

The machinery developed in Section 1 is used in Section 2 to prove the equivalence of various definitions of A(X) and to define the linearization transformation A(X)\(\to K({\mathbb{Z}}[GX]).\)

Section 3 is devoted to an abstract version (in the spirit of Section 1) of the relation of A(X) to concordance theory, represented by the Whitehead space \(Wh^{PL}(X)\). The author defines \(Wh^{PL}(X)\) as the realization of a simplicial category and establishes the homotopy fibration \[ h(X,A(*))\to A(X)\to Wh^{PL}(X) \] where h(X,A(*)) is the homology theory associated to A(*).

Reviewer: R.Vogt

##### MSC:

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

55N15 | Topological \(K\)-theory |

57Q60 | Cobordism and concordance in PL-topology |

55R50 | Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory |

18D30 | Fibered categories |