an:04190548
Zbl 0722.51003
Orlik, Peter
Introduction to arrangements
EN
Regional Conference Series in Mathematics 72. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0723-4). x, 110~p. (1989).
00408085
1989
b
51D99 52C35 51-02 05B35 51D25 06B25 57N65
matroid; arrangement of hyperplanes; survey; arrangements; lattices; M??bius function; Poincar?? polynomial of a lattice; differential forms; free arrangements; reflection arrangements
In the author's words: ``An arrangement of hyperplanes is a finite collection of codimension one subspaces in a finite dimensional vector space over some field. Arrangements occur in several branches of mathematics: in the study of braids and phase transition, in wave fronts, in hypergeometric functions, in reflection groups and Lie algebras, in coding theory, in the study of certain singularities, in combinatorics and group theory,and in spline functions.'' This survey of arrangements begins with a set of modern definitions and proceeds with a brief treatment of the combinatorial background required (lattices, the M??bius function, the Poincar?? polynomial of a lattice, etc.). Successive chapters deal with combinatorial algebras, lattice homology, the topology of the complement of an arrangement over complex numbers, the cohomology of the complement, the algebra of differential forms, recent developments in the study of the topology of the complement, free arrangements, and reflection arrangements. The treatment is highly topological rather than purely geometric, in contrast to the classical work of B. Gr??nbaum on arrangements. The present survey draws heavily on the work of V. I. Arnold, E. Brieskorn, P. Deligne, M. Falk, T. Kohno, K. Saito, H. Terao, T. Zaslavsky and the author's own research.
G.L.Alexanderson (Santa Clara)