Rational homotopy groups and Koszul algebras

Stefan Papadima; Alexander I. Suciu

doi:10.1016/S1631-073X(02)02420-2

Rational homotopy groups and Koszul algebras
[Groupes d'homotopie rationnels et algèbres de Koszul]

Stefan Papadima ¹ ; Alexander I. Suciu ²

¹ Institute of Mathematics of the Romanian Academy, PO Box 1-764, RO-70700 Bucharest, Romania
² Department of Mathematics, Northeastern University, Boston, MA 02115, USA

Comptes Rendus. Mathématique, Volume 335 (2002) no. 1, pp. 53-58.

Résumé
Abstract

Soient X et Y deux CW-espaces de type fini (X connexe, Y simplement connexe), tels que l'anneau de cohomologie $H^{*} (Y, ℚ)$ soit un k-recalibrage de $H^{*} (X, ℚ)$ . Si $H^{*} (X, ℚ)$ est une algèbre de Koszul, alors l'algèbre de Lie graduée $π_{*} (Ω Y) \otimes ℚ$ est le k-recalibrage de ${gr}_{*} (π_{1} X) \otimes ℚ$ . Si Y est un espace formel, alors l'implication réciproque est vraie aussi, et l'espace Y est coformel. De plus, si X est formel, avec algèbre de cohomologie de Koszul, on trouve des isomorphismes de groupes filtrés entre le complété de Malcev de π₁X, le complété de $[Ω S^{2 k + 1}, Ω Y]$ , et le groupe de Milnor–Moore d'applications de cogèbres entre $H_{*} (Ω S^{2 k + 1}, ℚ)$ et $H_{*} (Ω Y, ℚ)$ .

Let X and Y be finite-type CW-spaces (X connected, Y simply connected), such that the ring $H^{*} (Y, ℚ)$ is a k-rescaling of $H^{*} (X, ℚ)$ . If $H^{*} (X, ℚ)$ is a Koszul algebra, then the graded Lie algebra $π_{*} (Ω Y) \otimes ℚ$ is the k-rescaling of ${gr}_{*} (π_{1} X) \otimes ℚ$ . If Y is a formal space, then the converse holds, and Y is coformal. Furthermore, if X is formal, with Koszul cohomology algebra, there exist filtered group isomorphisms between the Malcev completion of π₁X, the completion of $[Ω S^{2 k + 1}, Ω Y]$ , and the Milnor–Moore group of coalgebra maps from $H_{*} (Ω S^{2 k + 1}, ℚ)$ to $H_{*} (Ω Y, ℚ)$ .

Reçu le : 2002-01-28
Accepté le : 2002-04-29
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02420-2

Affiliations des auteurs :

Stefan Papadima ¹ ; Alexander I. Suciu ²

¹ Institute of Mathematics of the Romanian Academy, PO Box 1-764, RO-70700 Bucharest, Romania
² Department of Mathematics, Northeastern University, Boston, MA 02115, USA

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Stefan Papadima; Alexander I. Suciu. Rational homotopy groups and Koszul algebras. Comptes Rendus. Mathématique, Volume 335 (2002) no. 1, pp. 53-58. doi : 10.1016/S1631-073X(02)02420-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02420-2/

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