Isomorphism conjectures in algebraic 𝐾-theory

Abstract

duff In this paper we are concerned with the four functors Y., Y3 if, *, and *, which map from the category of topological spaces X to the category diff of Q-spectra. The functor 3*( ) (or ?if ( )) maps the space X to the Qspectrum of stable topological (or smooth) pseudoisotopies of X. (The spaces diff dif 3i(X), id9 (X), i > 0, are Hatcher's deloopings of 970(X), % Yd(X) (cf. [29])). The functor J*( ) maps a path-connected space X to the algebraic Ktheoretic S2-spectrum for the integral group ring Z7r1X. (The spaces X(X), i > 0, are the Gersten-Wagoner deloopings of Z%O(X); cf. [27, 52].) 5*(X) is also defined on a nonpath-connected space X to be () (X1), where {Xi: i e I} are the path components of X and iI (Xi) indicates the direct limit of all finite products of the {I (Xi): i E I}. If X is path connected then the functor *-' ( ) maps X to the L -surgery classifying spaces for oriented surgery problems with fundamental group r I1X identified by their fourfold periodicity k 90(X) = k+4j (X) (cf. [35, 39, 44, 47]). In addition, if X is not path connected then we set Y77(X) = tiEI5*fJ'(Xi), where {Xi: i E i} are the path components of X. Results obtained by the authors over the past five years (cf. [15-25] and, in particular, [19; 16, A.1, A.18]), together with many earlier results and conjectures which are reviewed in 1.6.1-1.6.6, suggest that, for any connected CWcomplex X and for 5* () denoting any of the above functors, the Q-spectrum *(X) should be computable in a simple way from the values taken by 9'*() on covering spaces of X, which have very simple fundamental groups. In more detail, we let 9'(X) denote the collection of all subgroups H c ir IX, which are either finite or virtually infinite cyclic. A group H is virtually infinite cyclic if there is a short exact sequence of groups 1 -+ Z -+ H -+ F -+ 1 with F equal to a finite group and Z equal to the infinite cyclic group. For each such group we let XH -+ X denote the covering space projection corresponding to H. Then the Q-spectrum 5? (X) should be computable in a simple way from the Q-spectra {5 (XH): H E 9 (X)}. The purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any X