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Isomorphism conjectures in algebraic \(K\)-theory. (English) Zbl 0798.57018
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.

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
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