Geometry & Topology

Combinatorial Morse theory and minimality of hyperplane arrangements

Mario Salvetti and Simona Settepanella

Full-text: Open access


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.

Article information

Geom. Topol., Volume 11, Number 3 (2007), 1733-1766.

Received: 28 March 2007
Accepted: 18 July 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32S22: Relations with arrangements of hyperplanes [See also 52C35]
Secondary: 52C35: Arrangements of points, flats, hyperplanes [See also 32S22] 32S50: Topological aspects: Lefschetz theorems, topological classification, invariants

Morse theory arrangements combinatorics


Salvetti, Mario; Settepanella, Simona. Combinatorial Morse theory and minimality of hyperplane arrangements. Geom. Topol. 11 (2007), no. 3, 1733--1766. doi:10.2140/gt.2007.11.1733.

Export citation


  • A Bj örner, G M Ziegler, Combinatorial stratification of complex arrangements, J. Amer. Math. Soc. 5 (1992) 105–149
  • N Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles 1337, Hermann, Paris (1968)
  • D C Cohen, Cohomology and intersection cohomology of complex hyperplane arrangements, Adv. Math. 97 (1993) 231–266
  • D C Cohen, P Orlik, Arrangements and local systems, Math. Res. Lett. 7 (2000) 299–316
  • A Dimca, S Papadima, Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments, Ann. of Math. $(2)$ 158 (2003) 473–507
  • H Esnault, V Schechtman, E Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992) 557–561
  • R Forman, Morse theory for cell complexes, Adv. Math. 134 (1998) 90–145
  • R Forman, A user's guide to discrete Morse theory, Sém. Lothar. Combin. 48 (2002) Art. B48c, 35 pp.
  • D B Fuks, Cohomology of the braid group $\mathrm{mod}~2$, Funkcional. Anal. i Priložen. 4 (1970) 62–73
  • I M Gelfand, G L Rybnikov, Algebraic and topological invariants of oriented matroids, Dokl. Akad. Nauk SSSR 307 (1989) 791–795
  • T Kohno, Homology of a local system on the complement of hyperplanes, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986) 144–147
  • A Libgober, S Yuzvinsky, Cohomology of the Orlik–Solomon algebras and local systems, Compositio Math. 121 (2000) 337–361
  • P Orlik, H Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften 300, Springer, Berlin (1992)
  • R Randell, Morse theory, Milnor fibers and minimality of hyperplane arrangements, Proc. Amer. Math. Soc. 130 (2002) 2737–2743
  • M Salvetti, Topology of the complement of real hyperplanes in $\mathbb{C}^N$, Invent. Math. 88 (1987) 603–618
  • M Salvetti, The homotopy type of Artin groups, Math. Res. Lett. 1 (1994) 565–577
  • V Schechtman, H Terao, A Varchenko, Local systems over complements of hyperplanes and the Kac–Kazhdan conditions for singular vectors, J. Pure Appl. Algebra 100 (1995) 93–102
  • A I Suciu, Translated tori in the characteristic varieties of complex hyperplane arrangements, Topology Appl. 118 (2002) 209–223 Arrangements in Boston: a Conference on Hyperplane Arrangements (1999)
  • M Yoshinaga, Hyperplane arrangements and Lefschetz's hyperplane section theorem, to appear in Kodai Math. J. (2007)
  • T Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975) vii+102