The Segal Conjecture for Infinite Groups
arXiv:1901.09250 · 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 zero-th stable cohomotopy of the classifying space BG is isomorphic to the I-adic completion of the ring given by the zero-th equivariant stable cohomotopy of underline{E}G for I the augmentation ideal.
Final version: Algebraic & Geometric Topology 20-2 (2020), 965--986. DOI 10.2140/agt.2020.20.965