Volume 22, issue 6 (2018)

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On the Farrell–Jones conjecture for Waldhausen's $A$–theory

Nils-Edvin Enkelmann, Wolfgang Lück, Malte Pieper, Mark Ullmann and Christoph Winges

Geometry & Topology 22 (2018) 3321–3394
Abstract

We prove the Farrell–Jones conjecture for (nonconnective) $A$–theory with coefficients and finite wreath products for hyperbolic groups, $CAT\left(0\right)$–groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties such as passing to subgroups, colimits of direct systems of groups, finite direct products and finite free products. These results hold also for Whitehead spectra and spectra of stable pseudoisotopies in the topological, piecewise linear and smooth categories.

Keywords
Farrell–Jones conjecture, aspherical closed manifolds, $A$–theory, Whitehead spaces, spaces of stable pseudoisotopies, spaces of stable $h$–cobordisms
Mathematical Subject Classification 2010
Primary: 19D10
Secondary: 57Q10, 57Q60