Combinatorial Morse theory and minimality of hyperplane arrangements

Salvetti, Mario¹ ; Settepanella, Simona²

¹ Dipartimento di Matematica “L Tonelli", Università di Pisa, Largo B Pontecorvo 5, 56127 Pisa, Italy
² Dipartimento di Matematica “L Tonelli", Universitaà di Pisa, Largo B Pontecorvo 5, 56127 Pisa, Italy

Geometry & topology, Tome 11 (2007) no. 3, pp. 1733-1766.

Résumé

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.

Détail

DOI : 10.2140/gt.2007.11.1733

Keywords: Morse theory, arrangements, combinatorics

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