Homology, lower central series, and hyperplane arrangements
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Homology, lower central series, and hyperplane arrangements Richard D. Porter1
· Alexander I. Suciu1
To the memory of S¸ tefan Papadima, 1953–2018 Received: 13 June 2019 / Accepted: 28 November 2019 © Springer Nature Switzerland AG 2019
Abstract We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of Grigory Rybnikov in a more general framework and leads to an application to hyperplane arrangements, whereby we show that all the nilpotent quotients of a decomposable arrangement group are combinatorially determined. Keywords Lower central series · Tower of nilpotent quotients · Cohomology ring · Associated graded Lie algebra · Holonomy Lie algebra · Malcev Lie algebra · Hyperplane arrangement · Decomposable arrangement Mathematics Subject Classification 16S37 · 16W70 · 17B70 · 20F14 · 20F18 · 20F40 · 20J05 · 55P62 · 57M05
1 Introduction 1.1 Motivation The motivation for this paper comes from an effort to understand Rybnikov’s invariant used in [36–38] to distinguish between the fundamental groups of complements
A.I. Suciu partially supported by the Simons Foundation Collaboration Grant for Mathematicians # 354156.
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Richard D. Porter [email protected] Alexander I. Suciu [email protected]
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Department of Mathematics, Northeastern University, Boston, MA 02115, USA
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R.D. Porter, A.I. Suciu
of two hyperplane arrangements with the same incidence structure. Work of Arnol’d, Brieskorn, and Orlik–Solomon insures that an arrangement complement, M(A), is rationally formal, and that the cohomology ring H ∗ (M(A)) is determined solely by the intersection lattice, L(A). Thus, the complements of the Rybnikov pair of arrangements share the same rational homotopy type; in particular, the respective fundamental groups share the same rational associated graded Lie algebras and second nilpotent quotients. Nevertheless, the third nilpotent quotients of those two groups are not isomorphic, for reasons that are to this date somewhat mysterious, despite repeated attempts to elucidate this phenomenon, see e.g. [1,2,26,27]. We take here a different approach, closely modeled on Rybnikov’s original approach from [36,37], yet from a more general point of view. In the process, we develop a machinery for determining when a given isomorphism between the n-th nilpotent quotients of two groups satisfying certain mild finiteness and homological assumptions extends to an isomorphism between the (n + 1)-st stages of the respective nilpotent towers. 1.2 The holonomy map Let X be a connected CW-complex. We will assume throughout that the first homology group H1 (X ) is finitely generated and torsion-free. Let G = π1 (X ) be the fundamental group of X , and let n (G) denote its lower central series subgroups. Finally, let X → K (G ab , 1) be a classifying map corresponding to the abelianization homomorph
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