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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(0)–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
References
Publication
Received: 26 November 2016
Accepted: 4 March 2018
Published: 23 September 2018
Proposed: Steve Ferry
Seconded: Martin Bridson, Benson Farb
Authors
Nils-Edvin Enkelmann
Mathematisches Institut
Rheinische Wilhelms-Universität Bonn
Bonn
Germany
https://www.hcm.uni-bonn.de/people/profile/nils-edvin-enkelmann/
Wolfgang Lück
Mathematisches Institut
Rheinische Wilhelms-Universität Bonn
Bonn
Germany
https://www.him.uni-bonn.de/lueck/
Malte Pieper
Mathematisches Institut
Rheinische Wilhelms-Universität Bonn
Bonn
Germany
https://www.hcm.uni-bonn.de/people/profile/malte-pieper/
Mark Ullmann
Institut für Mathematik
FU Berlin
Berlin
Germany
Christoph Winges
Max-Planck-Institut für Mathematik
Bonn
Germany
http://people.mpim-bonn.mpg.de/winges/