Homotopy Lie algebras, lower central series and the Koszul property

Details Homotopy Lie algebras, lower central series and the Koszul property Homotopy Lie algebras, lower central series and the Koszul property

2001-10-28

arxiv.org/abs/math/0110303v2

Geom. Topol. 8 (2004), 1079-1125

Mathematics

Algebraic Topology

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper30.abs.html

Scientific paper

10.2140/gt.2004.8.1079

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.

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