Geometry & Topology, Vol. 8 (2004) Paper no. 30, pages 1079--1125.

Homotopy Lie algebras, lower central series and the Koszul property

Stefan Papadima, Alexander I Suciu

Abstract. Let X and Y be finite-type CW--complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie algebra associated to the lower central series of pi_1(X). If Y is a formal space, this equality is actually equivalent to the Koszulness of H^*(X,Q). If X is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of pi_1(X) and the completion of [Omega S^{2k+1} ,Omega Y]. Among spaces that admit naturally defined homological rescalings are complements of complex hyperplane arrangements, and complements of classical links. The Rescaling Formula holds for supersolvable arrangements, as well as for links with connected linking graph.

Keywords. Homotopy groups, Whitehead product, rescaling, Koszul algebra, lower central series, Quillen functors, Milnor--Moore group, Malcev completion, formal, coformal, subspace arrangement, spherical link

AMS subject classification. Primary: 16S37, 20F14, 55Q15. Secondary: 20F40, 52C35, 55P62, 57M25, 57Q45.

DOI: 10.2140/gt.2004.8.1079

E-print: arXiv:math.AT/0110303

Submitted to GT on 3 March 2004. Paper accepted 17 July 2004. Paper published 22 August 2004.

Notes on file formats

Stefan Papadima, Alexander I Suciu
Institute of Mathematics of the Romanian Academy
PO Box 1-764, RO-014700 Bucharest, Romania
Department of Mathematics, Northeastern University
Boston, MA 02115, USA


GT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to