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Combinatorial Morse
theory and minimality of hyperplane arrangements
Mario Salvetti and Simona Settepanella |
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Geometry & Topology 11 (2007) 1733–1766
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arXiv: 0705.2874 |
Abstract |
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Using combinatorial Morse theory on the CW–complex 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 , such that 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
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Mathematical Subject Classification 2000
Primary: 32S22
Secondary: 52C35, 32S50
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References |
Forward citations |
Publication
Received: 28 March 2007
Accepted: 18 July 2007
Published: 24 September 2007
Proposed: Walter Neumann
Seconded: Ralph Cohen, Tom Goodwillie
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Authors |
| Mario Salvetti | |
| Dipartimento di Matematica “L
Tonelli" Università di Pisa Largo B Pontecorvo 5 56127 Pisa Italy |
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| Simona Settepanella | |
| Dipartimento di Matematica “L
Tonelli" Universitaà di Pisa Largo B Pontecorvo 5 56127 Pisa Italy |
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