Farrell, F. T.; Jones, L. E. Isomorphism conjectures in algebraic \(K\)-theory. (English) Zbl 0798.57018 J. Am. Math. Soc. 6, No. 2, 249-297 (1993). This paper describes a conjecture on the calculation of four functors \({\mathcal P}_ *\), \({\mathcal P}_ *^{\text{diff}}\), \({\mathcal K}_ *\), and \({\mathcal L}_ *^{-\infty}\) that map the category of topological spaces to the category of \(\Omega\)-spectra. The functor \({\mathcal P}_ *\) (or \({\mathcal P}_ *^{\text{diff}}\)) maps the space \(X\) to the \(\Omega\)- spectrum of stable topological (or smooth) pseudoisotopies of \(X\). The functor \({\mathcal K}_ *\) maps the path-connected space \(X\) to the algebraic \(K\)-theoretic \(\Omega\)-spectrum for the integral group ring \(Z\pi_ 1 X\). If \(X\) is path connected then the functor \({\mathcal L}_ *^{-\infty}\) maps \(X\) to the \(L^{-\infty}\)-surgery classifying spaces for oriented surgery problems with fundamental group \(\pi_ 1\). Results obtained by the authors and others over recent years indicate that for each of these functors it should be possible to compute the associated \(\Omega\)-spectra from the \(\Omega\)-spectra associated to the covering spaces \(X_ H\) where \(H\) runs through the subgroups of \(\pi_ 1 X\) that are either finite or virtually infinite cyclic. The paper formulates a precise conjecture along these lines and verifies it for any \(X\) whose fundamental group is a co-compact discrete subgroup of a virtually connected Lie group in the case of the two functors \({\mathcal P}_ *\) and \({\mathcal P}_ *^{\text{diff}}\). A similar verification for the functor \({\mathcal L}_ *^{-\infty}\) is promised for a later paper. Reviewer: J.Hodgson (Philadelphia) Cited in 27 ReviewsCited in 80 Documents MSC: 57R67 Surgery obstructions, Wall groups 19D50 Computations of higher \(K\)-theory of rings 57N37 Isotopy and pseudo-isotopy 19D35 Negative \(K\)-theory, NK and Nil Keywords:stable topological pseudoisotopies; assembly map; surgery group; \(\Omega\)-spectra; \(L^{-\infty}\)-surgery classifying spaces; covering spaces; fundamental group; co-compact discrete subgroup of a virtually connected Lie group PDF BibTeX XML Cite \textit{F. T. Farrell} and \textit{L. E. Jones}, J. Am. Math. Soc. 6, No. 2, 249--297 (1993; Zbl 0798.57018) Full Text: DOI