The Minimal Model of the Complement of an Arrangement of Hyperplanes

Abstract

ABSTRACT. In this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory. Let A be a finite collection of hyperplanes in C', and let M = C' -UHeA H. We say A is a rational K(ir, 1) arrangement if the rational completion of M is aspherical. For these arrangements an identity (the LCS formula) is established relating the lower central series of tt\ (M) to the cohomology of M. This identity was established by group-theoretic means for the class of fiber-type arrangements in previous work. We reproduce this result by showing that the class of rational K(ir, 1) arrangements contains all fiber-type arrangements. This class includes the reflection arrangements of types A/ and B[. There is much interest in arrangements for which Af is a K(ir, 1) space. The methods developed here do not apply directly because M is rarely a nilpotent space. We give examples of K(ir, 1) arrangements which are not rational K(tt,1) for which the LCS formula fails, and K(tt, 1) arrangements which are not rational K(tt, 1) where the LCS formula holds. It remains an open question whether rational K(it, 1) arrangements are necessarily K(ir, 1).

Fundamental Groups of Real Arrangements and Torsion in the Lower Central Series Quotients

“…Concerning the cohomology algebra of M (A), Orlik and Solomon [OS80] showed that it is determined by L(A) by constructing a combinatorial algebra which is isomorphic to H • (M (A)), previously computed by Arnol'd and Brieskorn [Bri73] in some outstanding cases. Using Sullivan 1-minimal models, Falk proves that also φ k (G A ) is combinatorially determined [Fal88]. He also obtains with Randell [FR85], in the particular case of fibertype arrangements, the LCS formula:…”

Cohomology rings and nilpotent quotients of real and complex arrangements

“…We will define in the latter sections a series of invariants of both H ≤2 (X) and G/G 3 , and relate them one to another. For now, let us define invariants of χ, following an idea of Ziegler [34], that originated from Falk's work on minimal models of arrangements [10].…”

Rational Homotopy Theory via Sullivan Models: A Survey

“…The arrangement of hyperplanes is of fiber-type if there is a tower M = M n pn → M n−1 → · · · p 2 → M 1 = C * where each M k is the complement of an arrangement of hyperplanes in C k , and where p k is the restriction of a linear map C k → C k−1 whose fiber is a copy of C with finitely many points removed. In that case the minimal Sullivan model (∧V, d) of M satisfies V = V 1 ( [29]).…”