On the rational homotopy Lie algebra of spaces with finite dimensional rational cohomology and homotopy
Annales de l'Institut Fourier, Volume 39 (1989) no. 1, p. 193-206

The problem of the characterization of graded Lie algebras which admit a realization as the homotopy Lie algebra of a space of type F is discussed. The central results are formulated in terms of varieties of structure constants, several criterions for concrete algebras are also deduced.

On discute le problème de la caractérisation des algèbres de Lie graduées qui peuvent être réalisés comme des algèbres de Lie homotopiques d’espace de type F. Les résultats principaux sont exprimés à l’aide de la notion de variété des constantes structurales. On démontre aussi quelques critères pour des algèbres concrètes.

@article{AIF_1989__39_1_193_0,
     author = {Markl, Martin},
     title = {On the rational homotopy Lie algebra of spaces with finite dimensional rational cohomology and homotopy},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {39},
     number = {1},
     year = {1989},
     pages = {193-206},
     doi = {10.5802/aif.1163},
     zbl = {0657.55016},
     mrnumber = {90h:55018},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1989__39_1_193_0}
}
Markl, Martin. On the rational homotopy Lie algebra of spaces with finite dimensional rational cohomology and homotopy. Annales de l'Institut Fourier, Volume 39 (1989) no. 1, pp. 193-206. doi : 10.5802/aif.1163. http://www.numdam.org/item/AIF_1989__39_1_193_0/

[1] A. Borel, Linear algebraic groups, W.A. Benjamin, New-York, 1969. | MR 40 #4273 | Zbl 0186.33201

[2] J.-B. Friedlander, S. Halperin, An arithmetic characterization of the rational homotopy groups of certain spaces, Inv. Math., 53 (1979), 117-133. | MR 81f:55006b | Zbl 0396.55010

[3] S. Halperin, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc., 230 (1977), 173-199. | MR 57 #1493 | Zbl 0364.55014

[4] S. Halperin, Spaces whose rational homology and ѱ-homotopy are both finite dimensional, Astérisque, 113-114 (1984), 198-205. | MR 86a:55015 | Zbl 0546.55015

[5] S. Halperin, The structure of π*(ΩS), Astérisque, 113-114, 109-117. | MR 86b:55009 | Zbl 0546.55014

[6] R. Hartshorne, Algebraic geometry, Springer, 1977. | MR 57 #3116 | Zbl 0367.14001

[7] J.-M. Lemaire, F. Sigrist, Dénombrement de types d'homotopie rationnelle, C.R. Acad. Paris, Sér. A, 287 (1978), 109-112. | MR 80b:55009 | Zbl 0382.55005

[8] D. Quillen, Rational homotopy theory, Ann. Math., 90 (1969), 205-295. | MR 41 #2678 | Zbl 0191.53702

[9] P. Samuel, O. Zariski, Commutative algebra, Vol. I, Princeton N.J., Van Nostrand, 1958.

[10] P. Samuel, O. Zariski, Commutative algebra, Vol. II, Princeton N.J., Van Nostrand, 1960. | Zbl 0121.27801

[11] I.-R. Shafarevich, Osnovy algebraicheskoj geometrii, Moskva, 1972. | Zbl 0258.14001

[12] D. Tanré, Homotopie rationnelle : Modèles de Chen, Quillen, Sullivan, Lecture Notes in Math. 1025, Springer, 1983. | MR 86b:55010 | Zbl 0539.55001