An algebraic model for \(G\)-simple homotopy types
The authors' aim is to construct algebraic models for G-complexes X (G finite acting on X with nonvoid 1-connected fixed point sets \(X^ H\), for any subgroup H) which are adequate for the description of G-(simple) homotopy types and then to obtain classification results for G-(simple) homotopy types up to finite ambiguity, along the lines indicated by \textit{D. Sullivan} [Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 269-331 (1978; Zbl 0374.57002)] for ordinary homotopy types. The generalization is twofold: first, in the passage to the equivariant case, they obtain G- versions of results due to \textit{D. Quillen} [Ann. Math., II. Ser. 90, 205-295 (1969; Zbl 0191.537)] and \textit{H. J. Baues} and \textit{J. M. Lemaire} [Math. Ann. 225, 219-245 (1977; Zbl 0322.55019)] and establish the finiteness of the G-homotopy types with given G-DG Lie minimal model, integral structure and torsion bound. The same result was previously obtained by the second author [Astérisque 113/114, 312-337 (1984; see the previous review)] in terms of G-DG algebra models. The second point (motivating the Lie approach) is the construction of a G-DG Lie simple model for finite G-complexes, which is used in the place of the minimal Lie model in order to obtain the corresponding finiteness result for G- simple homotopy types. Finiteness results for G-(simple) homotopy types of \(Z_{p^ k}\) actions on spheres and of semilinear finite group actions are derived.
equivariant differential graded Lie algebras
finiteness result for G-simple homotopy types
scientific article; zbMATH DE number 3899745
scientific article