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**An algebraic model for G-homotopy types.**
*(English)*
Zbl 0564.55009

Homotopie algébrique et algèbre locale, Journ. Luminy/France 1982, Astérisque 113/114, 312-336 (1984).

[For the entire collection see Zbl 0535.00017.]

From the author’s introduction: ”Let G be a finite group. We consider simplicial complexes X on which G acts simplicially. We assume that all spaces in sight are G-nilpotent in the sense that the fixed point spaces \(X^ H\) are nonempty and nilpotent for all subgroups H of G. We also assume that the rational homology of each \(X^ H\) is of finite type. In [Trans. Am. Math. Soc. 274, 509-532 (1982; Zbl 0516.55010)] we constructed an algebraic invariant \({\mathcal M}_ X\) for a G-complex X, the equivariant minimal model of X, which generalizes D. Sullivan’s minimal model. The object of this paper is to use \({\mathcal M}_ X\) together with certain additional structure (lattices and torsion bounds) to classify G- complexes up to (integral) G-homotopy type.

Theorem 1.1: Let \({\mathcal M}\) be a minimal system of DGA’s with lattices Z and Z’ and let M be a positive integer. Then there are only finitely many finite G-complexes, up to G-homotopy type, of torsion bound M with equivariant minimal model \({\mathcal M}\) and lattices Z and Z’. The proof is based on the study of the group \(aut_ G(X)\) of G-homotopy classes of G- self homotopy equivalences of a G-space X. Let \(f: X\to X_ 0\) be an equivariant rationalization of X. Theorem 1.2: (i) The group \(aut_ G(X_ 0)\) is an algebraic Q-matrix group. (ii) \(aut_ G(X)\) is commensurable with an arithmetic subgroup of \(aut_ G(X_ 0)\). Hence \(aut_ G(X)\) is a finitely presented group. These generalize work by D. Sullivan in the nonequivariant case. The main arguments of Sullivan work equivariantly but they are technically much more delicate”. Since the proofs in Sullivan are very sketchy, the author provides complete details of the generalization.

From the author’s introduction: ”Let G be a finite group. We consider simplicial complexes X on which G acts simplicially. We assume that all spaces in sight are G-nilpotent in the sense that the fixed point spaces \(X^ H\) are nonempty and nilpotent for all subgroups H of G. We also assume that the rational homology of each \(X^ H\) is of finite type. In [Trans. Am. Math. Soc. 274, 509-532 (1982; Zbl 0516.55010)] we constructed an algebraic invariant \({\mathcal M}_ X\) for a G-complex X, the equivariant minimal model of X, which generalizes D. Sullivan’s minimal model. The object of this paper is to use \({\mathcal M}_ X\) together with certain additional structure (lattices and torsion bounds) to classify G- complexes up to (integral) G-homotopy type.

Theorem 1.1: Let \({\mathcal M}\) be a minimal system of DGA’s with lattices Z and Z’ and let M be a positive integer. Then there are only finitely many finite G-complexes, up to G-homotopy type, of torsion bound M with equivariant minimal model \({\mathcal M}\) and lattices Z and Z’. The proof is based on the study of the group \(aut_ G(X)\) of G-homotopy classes of G- self homotopy equivalences of a G-space X. Let \(f: X\to X_ 0\) be an equivariant rationalization of X. Theorem 1.2: (i) The group \(aut_ G(X_ 0)\) is an algebraic Q-matrix group. (ii) \(aut_ G(X)\) is commensurable with an arithmetic subgroup of \(aut_ G(X_ 0)\). Hence \(aut_ G(X)\) is a finitely presented group. These generalize work by D. Sullivan in the nonequivariant case. The main arguments of Sullivan work equivariantly but they are technically much more delicate”. Since the proofs in Sullivan are very sketchy, the author provides complete details of the generalization.

Reviewer: J.Stasheff

### MSC:

55P91 | Equivariant homotopy theory in algebraic topology |

55P62 | Rational homotopy theory |

57S17 | Finite transformation groups |