Combinatorial Morse theory and minimality of hyperplane arrangements

In [DP03], [Ra02] it was proven that the complement to a hyperplane arrangement in C is a minimal space, i.e. it has the homotopy type of a CW -complex with exactly as many i-cells as the i-th Betti number bi. The arguments use (relative) Morse theory and Lefschetz type theorems. This result of ”existence” was refined in the case of complexified real arrangements in [Yo05]. The author consider a flag V0 ⊂ V1 ⊂ · · · ⊂ Vn ⊂ R, dim(Vi) = i, which is generic with respect to the arrangement, i.e. Vi intersects transversally all codimensional−i intersections of hyperplanes. The interesting main result is a correspondence between the k−cells of the minimal complex and the set of chambers which intersect Vk but do not intersect Vk−1. The arguments still use the Morse theoretic proof of Lefschetz theorem, and some analysis of the critical cells is given. Unfortunately, the description does not allow to understand exactly the attaching maps of the cells of a minimal complex. In this paper we give, for a complexified real arrangement A, an explicit Lefschetz theorem free description of a minimal CW -complex. The idea is that, since an explicit CW−complex S which describes the homotopy type of the complement already exists (see [Sal87]), even if not minimal, one can work over such complex trying to ”minimize” it. A natural tool for doing that is to use combinatorial Morse theory over S. We follow the approach of [Fo98], [Fo02] to combinatorial Morse theory (i.e., Morse theory over CW -complexes). So, we explicitly construct a combinatorial gradient vector field over S, related to a given system of polar coordinates in R which is generic with respect to the arrangement A. Let S be the set of all facets of the stratification of R induced by the arrangement A (see [Bou68]). Then S has a natural partial ordering given by F ≺ G iff clos(F ) ⊃ G. Our definition of genericity of a coordinate system, which is stronger than that used in [Yo05], allows to give a total ordering ⊳ on S, which we call the polar ordering of the facets. The k-cells in S are in one-to-one correspondence with the pairs [C ≺ F ], where C is a chamber in S and F k is a codimensional-k facet of S which is contained in the closure of C. Then the gradient field can be recursively defined