Arrangements of hyperplanes. (English) Zbl 0757.55001

Orlik, Peter; Terao, Hiroaki

Grundlehren der Mathematischen Wissenschaften. 300. Berlin: Springer- Verlag. xviii, 325 p. (1992).

An arrangement of hyperplanes is a finite collection of codimension 1 affine subspaces in a finite dimensional vector space \(V\). This topic has given a lot of interesting results in the last 20 years using techniques from many different areas: algebraic topology, combinatorics, algebraic geometry, algebra, group actions; the main goal is usually to understand the “topology” of the complement of the hyperplanes in \(V\). In this very nice book the authors study arrangements with methods from all these areas. It is easy to follow the book (and learn a lot from it) even if one is not confident in some of these areas. Hence I recommend at least a glance to this book even to people not willing to do research on this topic: to learn many useful concepts and techniques and see how they may interact. For several years this should be “the” book on this topic; it is the first comprehensive study of arrangements of hyperplanes. I recommend it for self-study: it is essentially self-contained, provides proofs, stresses the foundations of the theory and arrives at the frontier of research, containing even new results (for non-central arrangements, i.e. when the intersection of all hyperplanes is empty).

Reviewer: E.Ballico (Povo)

MSC:

55-02	Research exposition (monographs, survey articles) pertaining to algebraic topology
55Q99	Homotopy groups
55P99	Homotopy theory
14F45	Topological properties in algebraic geometry
57N65	Algebraic topology of manifolds
32C18	Topology of analytic spaces

Keywords:

affine arrangement of hyperplanes; derivation; differential forms; logarithmic differential forms; Möbius function; Eilenberg MacLane space; arrangement of hyperplanes