Geometry & Topology

Homotopy Lie algebras, lower central series and the Koszul property

Ştefan Papadima and Alexander I Suciu

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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.

Article information

Geom. Topol., Volume 8, Number 3 (2004), 1079-1125.

Received: 3 March 2004
Accepted: 17 July 2004
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16S37: Quadratic and Koszul algebras 20F14: Derived series, central series, and generalizations 55Q15: Whitehead products and generalizations
Secondary: 20F40: Associated Lie structures 52C35: Arrangements of points, flats, hyperplanes [See also 32S22] 55P62: Rational homotopy theory 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}

homotopy groups Whitehead product rescaling Koszul algebra lower central series Quillen functors Milnor–Moore group Malcev completion formal coformal subspace arrangement spherical link


Papadima, Ştefan; Suciu, Alexander I. Homotopy Lie algebras, lower central series and the Koszul property. Geom. Topol. 8 (2004), no. 3, 1079--1125. doi:10.2140/gt.2004.8.1079.

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