Rational homotopy groups and Koszul algebras
[Groupes d'homotopie rationnels et algèbres de Koszul]
Comptes Rendus. Mathématique, Tome 335 (2002) no. 1, pp. 53-58.

Soient X et Y deux CW-espaces de type fini (X connexe, Y simplement connexe), tels que l'anneau de cohomologie H * (Y,) soit un k-recalibrage de H * (X,). Si H * (X,) est une algèbre de Koszul, alors l'algèbre de Lie graduée π * (ΩY) est le k-recalibrage de gr * (π 1 X). 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 [ΩS 2k+1 ,ΩY], et le groupe de Milnor–Moore d'applications de cogèbres entre H * (ΩS 2k+1 ,) et H * (ΩY,).

Let X and Y be finite-type CW-spaces (X connected, Y simply connected), such that the ring H * (Y,) is a k-rescaling of H * (X,). If H * (X,) is a Koszul algebra, then the graded Lie algebra π * (ΩY) is the k-rescaling of gr * (π 1 X). 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 [ΩS 2k+1 ,ΩY], and the Milnor–Moore group of coalgebra maps from H * (ΩS 2k+1 ,) to H * (ΩY,).

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DOI : 10.1016/S1631-073X(02)02420-2
Papadima, Stefan 1 ; Suciu, Alexander I. 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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Papadima, Stefan; Suciu, Alexander I. Rational homotopy groups and Koszul algebras. Comptes Rendus. Mathématique, Tome 335 (2002) no. 1, pp. 53-58. doi : 10.1016/S1631-073X(02)02420-2. http://www.numdam.org/articles/10.1016/S1631-073X(02)02420-2/

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