Isomorphism Conjectures in Algebraic K-Theory

F. T. Farrell, L. E. Jones
1993 Journal of The American Mathematical Society  
In this paper we are concerned with the four functors .9*, .9: iff , ~ , and £:-00 , which map from the category of topological spaces X to the category d'ff of n-spectra. The functor .9* ( ) (or .9* 1 ( )) maps the space X to the nspectrum of stable topological (or smooth) pseudoisotopies of X. (The spaces .9 i (X) , .9 i diff (X) , i ;::: 0, are Hatcher's deloopings of .9 0 (X) , .9~iff(X) (cf. [29])). The functor ~() maps a path-connected space X to the algebraic Ktheoretic n-spectrum for
more » ... integral group ring Z1l, X. (The spaces ~ (X) , i ;::: 0, are the Gersten-Wagoner deloopings of ~(X); cf. [27, 52] .) ~(X) is also defined on a nonpath-connected space X to be EBiEI~(Xi)' where {Xi: i E /} are the path components of X and EB iEI ~ (Xi) indicates the direct limit of all finite products of the {~(Xi): i E I}. If X is path connected then the functor £:-00 () maps X to the L -00 -surgery classifying spaces for oriented surgery problems with fundamental group 1l, X identified by their fourfold periodicity ~-OO(X) = ~~~(X) (cf. [35, 39, 44, 47] ). In addition, if X is not path connected then we set £:-OO(X) = EBiE1£:-00(X i ) , where {Xi: i E i} are the path components of X.
doi:10.2307/2152801 fatcat:whnp4oerdvegdjdakptxafvwge