The fundamental group of the complement of the complexification of a real arrangement of hyperplanes
type: Item
instance of: scholarly article
title: The fundamental group of the complement of the complexification of a real arrangement of hyperplanes
zbMATH Open document ID: 0921.55004
published in: Advances in Applied Mathematics
publication date: 1999-09-26T00:00:00Z
review text: Let \({\mathcal A}\) be a finite collection of lines in \({\mathbb{C}}P^2\). The homotopy type of the complement \(M={\mathbb{C}}P^2 - \bigcup {\mathcal A}\) is determined by the braid monodromy homomorphism \(\phi: F_{s} \to B_{n}\). This homomorphism is defined using a generic projection \({\mathbb{C}}P^{2} \to {\mathbb{C}}P^{1}\). The rank \(s\) of the free group \(F_{s}\) is the number of singular points of the curve \(\bigcup {\mathcal A}\). The monodromy of the projection along a loop \(x\) around the image of one of these singular points in \({\mathbb{C}}P^1\) yields a braid \(\phi(x)\) on \(n\) strands, \(n=| {\mathcal A}| \), realized as a self-diffeomorphism of \({\mathbb{C}}^2\). The fundamental group of \(M\) depends only on the braid monodromy group \(G\), the image of \(\phi\), considered as a subgroup of \(\Aut (F_n)\). Indeed, \(\pi_1(M)\) is the largest quotient of \(F_n\) on which \(G\) acts trivially. The paper under review is concerned with the dependence of \(G\) on the underlying combinatorial structure of \(\mathcal A\). The arrangement \(\mathcal A\) is assumed to be a complexified arrangement defined by real equations. Then the topology of \(M\) is determined by the oriented matroid of the real arrangement, via the associated Salvetti complex. The author defines a combinatorial braid monodromy group \(G_c\) using the Salvetti complex, and shows that it may replace \(G\) in the braid monodromy presentation of \(\pi_1(M).\) The oriented matroid, and the underlying matroid, are determined by a linear order of the lines of \(\mathcal A\). This order is assumed to be a ``realizable shelling order,'' determined by a generic line traversing the lines of the real arrangement. With this additional structure, it is asserted that \(G_c\), and thus the fundamental group of \(M\), depends only on the underlying unoriented matroid. The result applies to arrangements of arbitrary rank, since the fundamental group is determined by a 2-dimensional generic li…
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 1221606
zbMATH Keywords: arrangement
zbMATH Keywords: braid monodromy
zbMATH Keywords: oriented matroid
zbMATH Keywords: fundamental group
author: Raul Cordovil
reviewed by: Michael Falk
MaRDI profile type: Publication
full work available at URL: aama.1998.0611
OpenAlex ID: W2083020628
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DOI: 10.1006/AAMA.1998.0611
description: scientific article; zbMATH DE number 1221606