Combinatorial Morse theory and minimality of hyperplane arrangements

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.

💡 Research Summary

The paper addresses the long‑standing problem of proving the minimality of the complement of a complexified hyperplane arrangement by using purely combinatorial methods. The authors work with the Salvetti complex S, a regular CW‑model for the complement M = ℂⁿ \ ⋃_{H∈𝔄} H. Their main contribution is the explicit construction of a combinatorial gradient vector field on S, obtained from a total ordering of the facets of the real stratification induced by the arrangement. This ordering is produced by fixing a generic system of polar coordinates in ℝⁿ; each facet receives a pair (r,θ) consisting of its distance from the origin and its polar angle, and the facets are sorted lexicographically.

Using this order, the authors define a matching on the cells of S: a k‑cell σ is paired with a (k+1)‑cell τ whenever σ⊂τ and τ is the immediate successor of σ in the total order. Cells that cannot be matched become the critical (singular) facets of the Morse function. The paper gives a precise combinatorial description of these singular facets: they are exactly the facets that are minimal (or maximal) with respect to the polar order, and there is precisely one critical cell in each dimension. Consequently, after collapsing all matched pairs, the Morse complex reduces to a chain complex generated solely by the critical cells. This proves that the Salvetti complex, and hence the arrangement complement, is minimal: its cellular chain complex is homotopy equivalent to a complex with the smallest possible number of cells, namely the Betti numbers of the arrangement.

To compute the local homology at each critical facet, the authors introduce an algebraic chain complex L_* whose generators are the critical cells and whose boundary maps are dictated by the inclusion relations among the corresponding facets together with the polar order. They show that L_* is isomorphic to the cellular chain complex of S, so its homology coincides with H_*(M). In particular, the ranks of the critical cells agree with the dimensions of the Orlik–Solomon algebra, confirming the known relationship between Morse critical points and the arrangement’s cohomology.

A detailed case study is presented for the braid arrangement (type A_{n‑1}). Here the hyperplanes are given by x_i = x_j, and the polar coordinates are chosen so that the angle θ_{ij}=arg(x_i−x_j) provides a natural ordering θ_{12}<θ_{13}<…<θ_{n‑1,n}. The induced matching on the Salvetti complex yields critical cells that correspond exactly to the vertices of the Coxeter complex. The authors compute the resulting Morse complex explicitly and verify that it reproduces the known homology of the pure braid space, thereby illustrating the method in a concrete and highly symmetric setting.

Finally, the authors discuss the generality of their construction. Any arrangement that admits a generic polar system—essentially any arrangement after a small perturbation—can be treated in the same way. The polar order is combinatorial, and the matching algorithm can be implemented computationally, opening the way to algorithmic verification of minimality for large families of arrangements. Moreover, the approach provides a direct bridge between the Salvetti CW‑structure and the Orlik–Solomon algebra, offering new insight into the algebra‑topology correspondence for hyperplane arrangements.

In summary, the paper delivers a fully combinatorial proof of the minimality of complexified hyperplane arrangement complements. By constructing an explicit gradient vector field on the Salvetti complex via a polar‑coordinate induced total order, describing the singular facets, and presenting an algebraic complex that computes local homology, the authors not only recover known results for classical arrangements such as the braid arrangement but also lay out a framework that can be applied to arbitrary arrangements. This work enriches the toolkit of arrangement theory, replacing analytic Morse functions with discrete, algorithmically tractable structures while preserving the deep connections to Orlik–Solomon cohomology.