Fiber-type arrangements and factorization properties
Let \({\mathcal A}\) be an arrangement of hyperplanes in \({\mathbb{C}}^ n\), \(X={\mathbb{C}}^ n-\cup \{H|\) \(H\in {\mathcal A}\}\) and \({\mathcal G}_ X\) the holonomy Lie algebra over \({\mathbb{Q}}\) of X. In this paper, the author reviews the definitions and main properties of a fibre type arrangement and examines the broken-circuit algebra of an arrangement explaining its factorization property. The main result of the paper is the following theorem. Let x be a modular element of rank r(L(\({\mathcal A}))-1\), where L(\({\mathcal A})\) is the intersection lattice of \({\mathcal A}\), and Y be the base space of a fibration associated to x. Then the natural linear map \(\phi\) : Lib(A\({}_ 1)\oplus {\mathcal G}_ Y\to {\mathcal G}_ X\), where \(A_ 1=(\alpha_ 1,...,\alpha_{| A_ 1|})\), is an isomorphism of graded vector spaces. Finally, the author extends his results to affine arrangements.
lower central series of fundamental group
arrangement of hyperplanes in \({\mathbb{C}}^ n\)
scientific article; zbMATH DE number 4143039
scientific article