On G-disconnected injective models
Annales de l'Institut Fourier, Volume 53 (2003) no. 2, p. 625-664

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 HG. 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 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 HG.

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 HG. 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 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 HG.

DOI : https://doi.org/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
@article{AIF_2003__53_2_625_0,
     author = {Golasi\'nski, Marek},
     title = {On $G$-disconnected injective models},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {2},
     year = {2003},
     pages = {625-664},
     doi = {10.5802/aif.1954},
     zbl = {01940706},
     mrnumber = {1990008},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_2_625_0}
}
Golasiński, Marek. On $G$-disconnected injective models. Annales de l'Institut Fourier, Volume 53 (2003) no. 2, pp. 625-664. doi : 10.5802/aif.1954. http://www.numdam.org/item/AIF_2003__53_2_625_0/

[1] A.K. Bousfield; V.K.A.M. Gugenheim On PL de Rham theory and rational homotopy type, Memories of the Amer. Math. Soc., Tome 179 (1976) | MR 425956 | Zbl 0338.55008

[2] G. Bredon Equivariant Cohomology Theories, Lecture Notes in Math, Tome vol. 34 (1967) | MR 206946 | Zbl 0162.27202

[3] J.-M. Cordier; T. Porter Homotopy coherent category theory, Trans. Amer. Math. Soc., Tome 349 (1997) no. 1, pp. 1-54 | Article | MR 1376543 | Zbl 0865.18006

[4] P. Deligne; P. Griffiths; J. Morgan; D. Sullivan Real homotopy theory of Kähler manifolds, Invent. Math., Tome vol. 29 (1975), pp. 245-274 | Article | MR 382702 | Zbl 0312.55011

[5] B.L. Fine Disconnected equivariant rational homotopy theory and formality of compact G-Kähler manifolds (1992) (Ph.D. thesis, Chicago)

[6] B.L. Fine; G.V. Triantafillou On the equivariant formality of Kähler manifolds with finite group action, Can. J. Math., Tome 45 (1993), pp. 1200-1210 | Article | MR 1247542 | Zbl 0805.55009

[7] M. Golasiński Injective models of G-disconnected simplicial sets, Ann. Inst. Fourier, Grenoble, Tome 47 (1997) no. 5, pp. 1491-1522 | Article | Numdam | MR 1600367 | Zbl 0886.55012

[8] M. Golasiński Injectivity of functors to modules and DGA ' s, Comm. Algebra, Tome 27 (1999), pp. 4027-4038 | Article | MR 1700197 | Zbl 0942.18005

[9] M. Golasiński On the object-wise tensor product of functors to modules, Theory Appl. Categ., Tome 7 (2000) no. 1, pp. 227-235 | MR 1779432 | Zbl 0965.18008

[10] M. Golasiński Component-wise injective models of functors to DGAs, Colloq. Math., Tome 73 (1997), pp. 83-92 | MR 1436952 | Zbl 0877.55004

[11] S. Halperin Lectures on minimal models, Mémories S.M.F., Nouvelle série (1983), p. 9-10 | Numdam | Zbl 0536.55003

[12] P. Hilton; G. Mislin; J. Roitberg Localization of Nilpotent Groups and Spaces, Amsterdam, Noth-Holland Mathematics Studies, Tome 15 (1975) | MR 478146 | Zbl 0323.55016

[13] S. Lefschetz Algebraic Topology, Amer. Math. Soc. Colloq. Publ., Tome XXVII (1942) | Zbl 0061.39302

[14] D. Lehmann Théorie homotopique des formes différentielles, S.M.F., Astérique, Tome 45 (1977) | Zbl 0367.55008

[15] W. Lück Transformation groups and Algebraic K-Theory, Springer-Verlag, Lect. Notes in Math., Tome 1408 (1989) | MR 1027600 | Zbl 0679.57022

[16] J.P. May Simplicial Objects in Algebraic Topology, Princeton-Toronto-London-Melbourne, Van Nostrand Mathematical Studies, Tome 11 (1967) | MR 222892 | Zbl 0165.26004

[17] L.S. Scull Rational 𝕊 1 -equivariant homotopy theory, Trans. Amer. Math. Soc., Tome 354 (2002), pp. 1-45 | Article | MR 1859023 | Zbl 0989.55009

[18] D. Sullivan Infinitesimal Computations in Topology, Publ. Math. I.H.E.S., Tome 47 (1977), pp. 269-331 | Numdam | MR 646078 | Zbl 0374.57002

[19] G.V. Triantafillou Equivariant minimal models, Trans. Amer. Math. Soc., Tome 274 (1982), pp. 509-532 | Article | MR 675066 | Zbl 0516.55010

[20] G.V. Triantafillou An algebraic model for G-homotopy types, Astérisque, Tome 113-114 (1984), pp. 312-337 | MR 749073 | Zbl 0564.55009