Volume 11, issue 3 (2007)

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Combinatorial Morse theory and minimality of hyperplane arrangements

Mario Salvetti and Simona Settepanella

Geometry & Topology 11 (2007) 1733–1766
 arXiv: 0705.2874
Abstract

Using combinatorial Morse theory on the CW–complex $S$ constructed by Salvetti [Invent. Math. 88 (1987) 603–618] which gives the homotopy type of the complement to a complexified real arrangement of hyperplanes, we find an explicit combinatorial gradient vector field on $S$, such that $S$ contracts over a minimal CW–complex.

The existence of such minimal complex was proved before Dimca and Padadima [Ann. of Math. (2) 158 (2003) 473–507] and Randell [Proc. Amer. Math. Soc. 130 (2002) 2737–2743] and there exists also some description of it by Yoshinaga [Kodai Math. J. (2007)]. Our description seems much more explicit and allows to find also an algebraic complex computing local system cohomology, where the boundary operator is effectively computable.

Keywords
Morse theory, arrangements, combinatorics
Mathematical Subject Classification 2000
Primary: 32S22
Secondary: 52C35, 32S50