Hyperplane arrangement

A hyperplane arrangement is a finite collection of hyperplanes in a vector space. The combinatorics of the arrangement — which subsets of hyperplanes intersect, and in what dimensions — is encoded in the intersection lattice, and the topology of the complement (the space obtained by removing all the hyperplanes) is deeply connected to this combinatorial structure. The complement of a hyperplane arrangement is the classifying space for the associated Artin group.

Hyperplane arrangements appear across mathematics: in the study of reflection groups, where the reflecting hyperplanes of a Coxeter group form the canonical example; in singularity theory, where the Milnor fiber is constructed from an arrangement; and in combinatorics, where the chromatic polynomial of a graph is a special case of the characteristic polynomial of an arrangement.

A hyperplane arrangement is not a collection of walls but a map of the rooms between them. The geometry of the complement — the space that remains when the walls are removed — is the true object of study, and it is this geometry that gives rise to the Artin group, the Salvetti complex, and the deep connections between combinatorics and topology.