×

zbMATH — the first resource for mathematics

Introduction to arrangements. (English) Zbl 0722.51003
Regional Conference Series in Mathematics 72. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0723-4). x, 110 p. (1989).
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 Möbius function, the Poincaré 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. Grünbaum 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.

MSC:
51D99 Geometric closure systems
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
51-02 Research exposition (monographs, survey articles) pertaining to geometry
05B35 Combinatorial aspects of matroids and geometric lattices
51D25 Lattices of subspaces and geometric closure systems
06B25 Free lattices, projective lattices, word problems
57N65 Algebraic topology of manifolds
PDF BibTeX XML Cite