no. 2
On G-disconnected injective models
[Sur les G-modèles injectifs non connexess]
Golasiński, Marek1
1 Nicholas Copernicus University, Faculty of Mathematics and Computer Science, Chopina 12/18, 87-100 Toruń (Pologne)
Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 625-664.

Si G est un groupe fini, L.S. Scull a observé que la définition originale de la minimalité équivariante n’est pas correcte dans le cas G-connexe par suite d’une erreur concernant des propriétés algébriques. Dans le cas G-non connexe la catégorie des orbites 𝒪(G) a été remplacée par la catégorie 𝒪(G,X), avec un objet pour chaque composante des sous-ensembles simpliciaux de points fixes X H d’un ensemble G-simplicial X, pour tous les sous-groupes H⊆G. Nous redéfinissons la minimalité équivariante et nous redéveloppons des résultats d’homotopie rationnelle pour les ensembles G-simpliciaux non connexes. Pour montrer l’existence d’un modèle minimal injectif ℳ X pour un ensemble G-simplicial X non connexe, nous remplaçons 𝒪(G,X) par la catégorie plus subtile 𝒪 ˜(G,X) avec un objet pour chaque 0-simplexe de sous-ensembles simpliciaux de points fixes X H , par tous les sous- groupes H⊆G.

Let G be a finite group. It was observed by L.S. Scull that the original definition of the equivariant minimality in the G-connected case is incorrect because of an error concerning algebraic properties. In the G-disconnected case the orbit category 𝒪(G) was originally replaced by the category 𝒪(G,X) with one object for each component of each fixed point simplicial subsets X H of a G-simplicial set X, for all subgroups H⊆G. We redefine the equivariant minimality and redevelop some results on the rational homotopy theory of disconnected G-simplicial sets. To show an existence of the injective minimal model ℳ X for a disconnected G-simplicial set X we replace 𝒪(G,X) by the more subtle category 𝒪 ˜(G,X) with one object for each 0-simplex of fixed point simplicial subsets X H , for all subgroups H⊆G.

DOI : 10.5802/aif.1954
Classification : 55P62, 55P91, 16W80, 18G30
Keywords: differential graded algebra, de Rham algebra, $EI$-category, $i$-elementary extension, $i$-minimal model, linearly compact (complete) $k$-module, Postnikov tower, quasi-isomorphism, rationalization, $G$-simplicial set
Mot clés : algèbre gradué différentiel, algèbre de de Rham, $EI$-catégorie, $i$-extension élémentaire, $i$-modèle minimal, $k$-modèle compact linéairement, tour de Postnikov, quasi-isomorphisme, rationalisation, ensemble simpliciel $G$

Golasiński, Marek 1

1 Nicholas Copernicus University, Faculty of Mathematics and Computer Science, Chopina 12/18, 87-100 Toruń (Pologne) 
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Golasiński, Marek. On $G$-disconnected injective models. Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 625-664. doi : 10.5802/aif.1954. https://aif.centre-mersenne.org/articles/10.5802/aif.1954/

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