The Segal conjecture for infinite discrete groups

Lück, Wolfgang

doi:10.2140/agt.2020.20.965

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The Segal conjecture for infinite discrete groups

Wolfgang Lück

Algebraic & Geometric Topology 20 (2020) 965–986

DOI: 10.2140/agt.2020.20.965

Abstract

We formulate and prove a version of the Segal conjecture for infinite groups. For finite groups it reduces to the original version. The condition that $G$ is finite is replaced in our setting by the assumption that there exists a finite model for the classifying space $\underline{E} G$ for proper actions. This assumption is satisfied for instance for word hyperbolic groups or cocompact discrete subgroups of Lie groups with finitely many path components. As a consequence we get for such groups $G$ that the zeroth stable cohomotopy of the classifying space $B G$ is isomorphic to the $I$ –adic completion of the ring given by the zeroth equivariant stable cohomotopy of $\underline{E} G$ for $I$ the augmentation ideal.

Keywords

equivariant cohomotopy, Segal conjecture for infinite discrete groups

Mathematical Subject Classification 2010

Primary: 55P91

References

Publication

Received: 26 January 2019

Revised: 5 July 2019

Accepted: 9 August 2019

Published: 23 April 2020

Authors

Wolfgang Lück
Mathematisches Institut Rheinische Wilhelms-Universität Bonn Bonn Germany
http://www.him.uni-bonn.de/lueck

Volume 20, issue 2 (2020)