Homotopy Lie algebras, lower central series and the Koszul property

Papadima, Ştefan; Suciu, Alexander I

doi:10.2140/gt.2004.8.1079

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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

DOI: 10.2140/gt.2004.8.1079

arXiv: math.AT/0110303

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 $π_{*} (Ω Y) \otimes ℚ$ equals, up to $k$ –rescaling, the graded rational Lie algebra associated to the lower central series of $π_{1} (X)$ . 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 $π_{1} (X)$ and the completion of $[Ω S^{2 k + 1}, Ω Y]$ . 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

Mathematical Subject Classification 2000

Primary: 16S37, 20F14, 55Q15

Secondary: 20F40, 52C35, 55P62, 57M25, 57Q45

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Publication

Received: 3 March 2004

Accepted: 17 July 2004

Published: 22 August 2004

Proposed: Haynes Miller

Seconded: Thomas Goodwillie, Steven Ferry

Authors

Ştefan Papadima
Institute of Mathematics of the Romanian Academy PO Box 1-764 RO-014700 Bucharest Romania
http://www.imar.ro/~spapadim/
Alexander I Suciu
Department of Mathematics Northeastern University Boston Massachusetts 02115 USA
http://www.math.neu.edu/~suciu/

Volume 8, issue 3 (2004)