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Comptes Rendus Mathématique
Volume 335, n° 1
pages 53-58 (2002)
Doi : S1631-073X(02)02420-2
Received : 28 January 2002 ;  accepted : 29 April 2002
Rational homotopy groups and Koszul algebras
Groupes d'homotopie rationnels et algèbres de Koszul
 

Stefan Papadima a , Alexander I. Suciu b
a Institute of Mathematics of the Romanian Academy, PO Box 1-764, RO-70700 Bucharest, Romania 
b Department of Mathematics, Northeastern University, Boston, MA 02115, USA 

Note presented by Jean-Pierre Serre

Abstract

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(ΩY)Q is the k -rescaling of gr(π1X)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 π 1 X , the completion of [ΩS2k+1,ΩY], and the Milnor-Moore group of coalgebra maps from H(ΩS2k+1,Q) to H(ΩY,Q). To cite this article: S. Papadima, A.I. Suciu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 53-58.

The full text of this article is available in PDF format.
Résumé

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(ΩY)Q est le k -recalibrage de gr(π1X)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 π 1 X , le complété de [ΩS2k+1,ΩY], et le groupe de Milnor-Moore d'applications de cogèbres entre H(ΩS2k+1,Q) et H(ΩY,Q). Pour citer cet article : S. Papadima, A.I. Suciu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 53-58.

The full text of this article is available in PDF format.


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