The Segal conjecture for infinite discrete groups

Lück, Wolfgang¹

¹Mathematisches Institut, Rheinische Wilhelms-Universität Bonn, Bonn, Germany

Algebraic and Geometric Topology, Tome 20 (2020) no. 2, pp. 965-986

Résumé

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 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 BG is isomorphic to the I–adic completion of the ring given by the zeroth equivariant stable cohomotopy of E¯G for I the augmentation ideal.

Détail

DOI : 10.2140/agt.2020.20.965

Classification : 55P91
Keywords: equivariant cohomotopy, Segal conjecture for infinite discrete groups

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