Arrangementsformal

Arrangementsformal is a term used informally to describe the formal study of mathematical arrangements, especially hyperplane arrangements, and the role of formality in their algebraic and topological properties. In this setting, an arrangement usually means a finite collection of hyperplanes in a vector space, with attention to the combinatorial data captured by its intersection lattice. A central algebraic object is the Orlik–Solomon algebra, which presents the cohomology ring of the arrangement’s complement and encodes the combinatorial structure of how the hyperplanes intersect.

Formality is a concept from rational homotopy theory. A space is formal if its rational homotopy type is determined by its cohomology algebra. For many complex hyperplane arrangement complements, the space is formal, meaning that higher-order topological information can be deduced from cohomology alone. This aligns the combinatorial data of the arrangement with topological invariants and has implications for the study of configuration spaces and braid groups.

Researchers study which arrangements have formal complements and how changes in the intersection lattice influence formality. The term arrangementsformal thus encompasses both the combinatorial analysis of arrangements and the algebraic-topological notion of formality, highlighting the intersection between discrete structure and continuous invariants.

See also: hyperplane arrangement, Orlik–Solomon algebra, rational homotopy theory, formality.