# zbMATH — the first resource for mathematics

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
Full Text: