Injective models of G-disconnected simplicial sets
Annales de l'Institut Fourier, Tome 47 (1997) no. 5, pp. 1491-1522.

Nous généralisons les résultats de G.V. Triantafillou et B. Fine sur les ensembles simpliciaux G-non connexes. On présente l’existence d’un modèle injectif minimal pour une 𝕀-algèbre complète, où 𝕀 est une EI-catégorie. Ensuite, nous utilisons la EI-catégorie 𝒪(G,X) associée à un G-ensemble simplicial X, pour appliquer ces résultats à la catégorie des G-ensembles simpliciaux.

Enfin, nous décrivons le G-type d’homotopie rationnelle d’un G-ensemble simplicial nilpotent en utilisant leur modèle injectif minimal

We generalize the results by G.V. Triantafillou and B. Fine on G-disconnected simplicial sets. An existence of an injective minimal model for a complete 𝕀-algebra is presented, for any EI-category 𝕀. We then make use of the EI-category 𝒪(G,X) associated with a G-simplicial set X to apply these results to the category of G-simplicial sets.

Finally, we describe the rational homotopy type of a nilpotent G-simplicial set by means of its injective minimal model.

@article{AIF_1997__47_5_1491_0,
     author = {Golasi\'nski, Marek},
     title = {Injective models of $G$-disconnected simplicial sets},
     journal = {Annales de l'Institut Fourier},
     pages = {1491--1522},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {5},
     year = {1997},
     doi = {10.5802/aif.1607},
     mrnumber = {99b:55020},
     zbl = {0886.55012},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1607/}
}
TY  - JOUR
AU  - Golasiński, Marek
TI  - Injective models of $G$-disconnected simplicial sets
JO  - Annales de l'Institut Fourier
PY  - 1997
SP  - 1491
EP  - 1522
VL  - 47
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://www.numdam.org/articles/10.5802/aif.1607/
DO  - 10.5802/aif.1607
LA  - en
ID  - AIF_1997__47_5_1491_0
ER  - 
%0 Journal Article
%A Golasiński, Marek
%T Injective models of $G$-disconnected simplicial sets
%J Annales de l'Institut Fourier
%D 1997
%P 1491-1522
%V 47
%N 5
%I Association des Annales de l’institut Fourier
%U https://www.numdam.org/articles/10.5802/aif.1607/
%R 10.5802/aif.1607
%G en
%F AIF_1997__47_5_1491_0
Golasiński, Marek. Injective models of $G$-disconnected simplicial sets. Annales de l'Institut Fourier, Tome 47 (1997) no. 5, pp. 1491-1522. doi : 10.5802/aif.1607. https://www.numdam.org/articles/10.5802/aif.1607/

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

[2] G.E. Bredon, Equivariant Cohomology Theories, Lecture Notes in Math., Springer-Verlag, 34 (1967). | MR | Zbl

[3] A.D. Elmendorf, System of fixed point sets, Trans. Amer. Math. Soc., 277 (1983), 275-284. | MR | Zbl

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

[5] B.L. Fine and G.V. Triantafillou, On the equivariant formality of Kähler manifolds with finite group action, Can. J. Math., 45 (1993), 1200-1210. | MR | Zbl

[6] M. Golasiński, Injectivity of the de Rham algebra on G-disconnected simplicial sets, (submitted).

[7] M. Golasiński, Equivariant Rational Homotopy Theory as a Closed Model Category, J. Pure Appl. Alg., (to appear). | Zbl

[8] M. Golasiński, Componentwise injective models of functors to DGAs, Colloq. Math., 73 (1997), 83-92. | MR | Zbl

[9] S. Halperin, Lectures on minimal models, Mémories S.M.F., nouvelle série, 9-10 (1983). | Numdam | MR | Zbl

[10] S. Lefschetz, Algebraic topology, Amer. Math. Soc. Colloq. Publ., XXVII (1942). | MR | Zbl

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

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

[13] G.V. Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc., 274 (1982), 509-532. | MR | Zbl

[14] G.V. Triantafillou, Ratinalization of Hopf G-spaces, Math. Z., 182 (1983), 485-500. | MR | Zbl

[15] G.V. Triantafillou, An algebraic model for G-homotopy types, Astérisque, 113-114 (1984), 312-337. | Numdam | MR | Zbl

Cité par Sources :