2026/07/22 by Ulrich Pennig, George Tridimas
Mathematics · #math.AT #math.OA
Lifting obstructions for group actions, cocycle actions, and Γ-kernels admit a cohomological description via topological crossed modules, as recently developed by Izumi, Giron-Pacheco, and the first named author. For a strongly self-absorbing C^*-algebra A, we show that the classifying spaces BDGA and BDPGA of the respective crossed modules governing cocycle actions and Γ-kernels, respectively, carry infinite loop space structures induced by the tensor product; the same holds for the crossed module BDGA whenever U(A) is connected. This confirms a conjecture from the aforementioned work and extends to Γ-kernels and cocycle actions the connection with stable homotopy theory. For the proof we construct \mathbbI-FCPs from the relevant crossed modules, pass to Γ-spaces, and apply the May-Thomason infinite loop space machine. Consequently, the natural transformation H1(Γ,G) → [BΓ,BDG] takes values in cohomology groups.