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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm167, 39 pages, doi:10.1093/imrn/rnm167 published on February 8, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Shelling-Type Orderings of Regular CW-Complexes and Acyclic Matchings of the Salvetti Complex

Emanuele Delucchi

The Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720-5070, USA

Correspondence: Correspondence to be sent to: delucchi{at}math.binghamton.edu

Motivated by the work of Salvetti and Settepanella [24, Combinatorial Morse theory and minimality of hyperplane arrangements, Remark 4.5], we give a purely combinatorial description of a class of discrete Morse functions having a minimal number of critical cells for the Salvetti complex of any linear arrangement. We start by studying certain total orderings of the cells of shellable regular CW-complexes, and use them to construct maximum acyclic matchings of the given complex. We apply this technique to the classical zonotope shellings. A new combinatorial stratification of the Salvetti complex allows us to paste such matchings and describe a class of maximum acyclic matchings of the whole complex. The construction can be done, so that the critical cells can be constructed from the chambers via the nbc sets. The results hold for abstract oriented matroids.


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