Group-theoretic algebraic models for homotopy types

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In this paper a nonabelian version of the Dold-Kan-Puppe theorem is provided, showing how the Moore-complex functor defines a full equivalence between the category of simplicial groups and the category of what is called 'hypercrossed complexes of groups', i.e. chain complexes of nonabelian groups (Gn,δn) with an additional structure in the form of binary operations G1 × G1 → Gk. We associate to a pointed topological space X a hypercrossed complex {A figure is presented}(X); and the functor {A figure is presented} induces an equivalence between the homotopy category of connected CW-complexes and a localization of the category of hypercrossed complexes. The relationship between {A figure is presented}(X) and Whitehead's crossed complex II(X) is established by a canonical surjection p:{A figure is presented}(X) → II(X), which is a quasi-isomorphism if and only if X is a J-complex. Algebraic models consisting of truncated chain-complexes with binary operations are deduced for n-types, and as an application we deduce a group-theoretic interpretation of the cohomology groups Hn(G, A). © 1991.




Carrasco, P., & Cegarra, A. M. (1991). Group-theoretic algebraic models for homotopy types. Journal of Pure and Applied Algebra, 75(3), 195–235.

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