Nonabelian algebraic topology
Connected homotopy \(n\)-types (that is, connected spaces with trivial homotopy groups \(\pi_i X\) for \(i>n\)) admit various algebraic models. More precisely, one wants a suitably algebraic category \(C\) such that some localization of \(C\) is equivalent to the homotopy category of connected homotopy \(n\)-types. For example, connected homotopy \(1\)-types are modeled by groups, and connected homotopy \(2\)-types are modeled by crossed modules, going back to Whitehead. The paper under review is a survey of models for connected homotopy \((n+1)\)-types. The author describes the functors relating crossed \(n\)-cubes of groups, \(n\)-cubes of fibrations, simplicial groups with Moore complex trivial above degree \(n\), and connected \((n+1)\)-types [\textit{J. Loday}, J. Pure Appl. Algebra 24, 179--202 (1982; Zbl 0491.55004), \textit{G. Ellis} and \textit{R. Steiner}, ibid. 46, 117--136 (1987; Zbl 0622.55010), \textit{T. Porter}, Topology 32, No. 1, 5--24 (1993; Zbl 0776.55012)]. He also describes a higher dimensional van Kampen theorem [\textit{R. Brown} and \textit{J. Loday}, ibid. 26, 311--335 (1987; Zbl 0622.55009)]. Section 5 surveys some applications: generalizations of the Blakers-Massey theorem, the homotopy excision theorem, the Hurewicz theorem, and the Hopf formula in group homology, as well as computing certain homotopy colimits of classifying spaces of groups. The paper is expository and contains no new results, except an explicit description of the coproduct of crossed \(n\)-cubes of groups.
scientific article; zbMATH DE number 6485539
scientific article