GROUP-THEORETIC ALGEBRAIC MODELS FOR HOMOTOPY TYPES

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 (G(n), delta-n) with an additional structure in the form of binary operations G(i) x G(j) --> G(k). We associate to a pointed topological space X a hypercrossed complex v(X); and the functor v induces an equivalence between the homotopy category of connected CW-complexes and a localization of the category of hypercrossed complexes. The relationship between v(X) and Whitehead's crossed complex PI(X) is established by a canonical surjection p: v(X) --> PI(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 H(n)(G,A).