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Combinatorial Morse theory and minimality of hyperplane arrangements. (English) Zbl 1134.32010

The complement to a hyperplane arrangement in \(\mathbb C^n\) is a minimal space; it has the homotopy type of a CW-complex with exactly as many \(i\)-cells as the \(i\)th Betti number. This was proven by A. Dimca and S. Papadima [Ann. Math. (2) 158, No. 2, 473–507 (2003; Zbl 1068.32019)] and R. Randell [Proc. Am. Math. Soc. 130, No. 9, 2737–2743 (2002; Zbl 1004.32010)] using (relative) Morse theory and Lefschetz type theorems. M. Yoshinaga [Kodai Math. J. 30, No. 2, 157–194 (2007; Zbl 1142.32012)] refined this result in the case of complexified real arrangements, still using the Morse theoretic proof of the Lefschetz theorem.
In the paper under review, the authors give, for a complexified real arrangement, an explicit description of a minimal CW-complex which does not use the Lefschetz theorem.
Using combinatorial Morse theory on the CW-complex \(S\) constructed in M. Salvetti [Invent. Math. 88, 603–618 (1987; Zbl 0594.57009)] which gives the homotopy type of the complement to a complexified real arrangement of hyperplanes, the authors find an explicit combinatorial gradient vector field on \(S\), such that \(S\) contracts over a minimal CW-complex.
This construction also gives an explicit algebraic complex which computes local system cohomology of the complement space.
The authors find a generic polar ordering on the braid arrangement. They give a description of the complex \(S\) in this case in terms of tableaux of a special kind. They characterize the singular tableaux and find an algorithm to compare two tableaux with respect to the polar ordering.

MSC:

32S22 Relations with arrangements of hyperplanes
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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References:

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