Digital Object Identifier (DOI) 10.1007/s002080000120 Math. Ann. 318, 277–298 (2000) Mathematische Annalen Dold-Kan type theorem for Γ -groups Teimuraz Pirashvili Received March 1, 1999 / Revised January 5, 2000 / Published online July 20, 2000 – c  Springer-Verlag 2000 0. Introduction By a well-known theorem of Dold-Kan the Moore normalization establishes an equivalence between the category of simplicial abelian groups and the category ofchaincomplexes(see[6]).Ourmainresultshowsthatthereisasimilartheorem for Γ -groups. Here Γ is the category of finite based sets. For a functor F : Γ → Groups ,weconstructafunctor cr(F) : Ω → Groups ,where Ω isthecategory of all nonempty finite sets and surjections. This construction is based on the notion of cross-effects of functors [4], which is a generalization of the classical definition of Eilenberg and Mac Lane [8] to the non-abelian setup. Our version of the Dold-Kan theorem in the abelian case claims that the category of abelian Γ -groups is equivalent to the category of functors Ab Ω and the equivalence is given by F  → cr(F). In the non-abelian case one needs to consider functors T : Ω → Gr together with maps {, }: T n × T m → T n+m satisfying additional relations. Here T n is the value of T on the set n := {1, ··· ,n}. A Γ -group F is called polynomial if cr(F) n = 0 for n >> 0. It is a consequence of our Dold-Kan type theorem for Γ -groups, that up to obvious induction one can reduce the study of polynimial Γ -groups to the abelian case. As a sample application of this principle we prove that for any finite simpli- cial set K and for any polynomial Γ -group F one has π n (F(K)) = 0 for all n>(dimK)(degF). Our interest in Γ -groups comes from the famous result of Segal [16], who proved that Γ -spaces are combinatorial models for connective spectra (see also [1], [5]). Based on the Kan-Thurston theorem we show that any connective spec- trum can be obtained from a discrete Γ -group up to stable weak equivalence. Our next goal is to get information about homotopy of spectra corresponding to T. Pirashvili A.M. Razmadze Math. Inst.Alexidze str. 1, Tbilisi, 380093, Republic of Georgia