#### 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

and

Department of Mathematics,
Northeastern University

Boston, MA 02115, USA

Email: stefan.papadima@imar.ro, a.suciu@neu.edu

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