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Homotopy Lie algebras, lower central series and the Koszul property

Ştefan Papadima and Alexander I Suciu

Geometry & Topology 8 (2004) 1079–1125

arXiv: math.AT/0110303


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, ) is a Koszul algebra. Then, the homotopy Lie algebra π(ΩY ) equals, up to k–rescaling, the graded rational Lie algebra associated to the lower central series of π1(X). If Y is a formal space, this equality is actually equivalent to the Koszulness of H(X, ). If X is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of π1(X) and the completion of [ΩS2k+1,ΩY ]. 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.

homotopy groups, Whitehead product, rescaling, Koszul algebra, lower central series, Quillen functors, Milnor–Moore group, Malcev completion, formal, coformal, subspace arrangement, spherical link
Mathematical Subject Classification 2000
Primary: 16S37, 20F14, 55Q15
Secondary: 20F40, 52C35, 55P62, 57M25, 57Q45
Forward citations
Received: 3 March 2004
Accepted: 17 July 2004
Published: 22 August 2004
Proposed: Haynes Miller
Seconded: Thomas Goodwillie, Steven Ferry
Ştefan Papadima
Institute of Mathematics of the Romanian Academy
PO Box 1-764
RO-014700 Bucharest
Alexander I Suciu
Department of Mathematics
Northeastern University
Massachusetts 02115