Combinatorial Morse theory and minimality of hyperplane arrangements

Abstract

We find an explicit combinatorial gradient vector field on the well known complex S (Salvetti complex) which models the complement to an arrangement of complexified hyperplanes. The argument uses a total ordering on the facets of the stratification of R^n associated to the arrangement, which is induced by a generic system of polar coordinates. We give a combinatorial description of the singular facets, finding also an algebraic complex which computes local homology. We also give a precise construction in the case of the braid arrangement.

29 pages

Cited by in corpus (13)

• Chamber basis of the Orlik-Solomon algebra and Aomoto complex
• Euclidean matchings and minimality of hyperplane arrangements
• Homology graph of real arrangements and monodromy of Milnor Fiber
• Minimality of hyperplane arrangements and basis of local system cohomology
• Discrete Morse theory for moment-angle complexes of pairs (D^n,S^{n-1})
• Combinatorial stratifications and minimality of 2-arrangements
• Combinatorial polar orderings and recursively orderable arrangements
• Factorization statistics and bug-eyed configuration spaces
• Complements of hyperplane arrangements as posets of spaces
• Some computations on the characteristic variety of a line arrangement
• Semi-algebraic partition and basis of Borel-Moore homology of hyperplane arrangements
• The $\text{nbc}$ minimal complex of supersolvable arrangements
• An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs