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

Abstract

Abstract. Motivated by the work of Salvetti and Settepanella ([24, Remark 4.5]) we introduce certain total orderings of the faces of any shellable regular CW-complex (called shelling-type orderings) that can be used to explicitly construct maximum acyclic matchings of the poset of cells of the given complex. Building on an application of this method to the classical zonotope shellings (i.e., those arising from linear extensions of the tope poset) we describe a class of maximum acyclic matchings for the Salvetti complex of a linear complexified arrangement. To do this, we introduce and study a new purely combinatorial stratification of the Salvetti complex. For the obtained acyclic matchings we give an explicit description of the critical cells that depends only on the chosen linear extension of the poset of regions. It is always possible to choose the linear extension so that the critical cells can be explicitly constructed from the chambers of the arrangement via the bijection to no-broken-circuit sets defined by Jewell and Orlik [17]. Our method generalizes naturally to abstract oriented matroids.

The Morse complex of a line arrangement

“…Even if such formulas can be computed explicitly in a finite number of steps, our present boundary operators are much simpler; our construction can be fruitfully used in other cases, for example to explicitly compute the so called characteristic varieties of the arrangement (see for example [Su02,D07] for several references about characteristic varieties and [D08,DS07] for related constructions).…”

(Discrete) Morse Theory on Configuration Spaces

“…The analog result for d = 2 was found in [12] (see also [16], [3]), after the proof that the complement to the arrangement is a minimal space ( [5,14]). For d > 2 the configuration spaces are simply-connected, so by general results they have the homotopy type of a minimal CW -complex.…”

Minimality of toric arrangements

“…Every summand on the right hand side counts the number of generators in top degree cohomology or -equivalently -the number of top dimesional cells of a minimal CWmodel of the complement of the complexification of A [Y ]. By [13,Lemma 4.18 and Proposition 2] these top dimensional cells correspond bijectively to chambers…”

“…In the case of complexified arrangements, explicit constructions of a minimal CWcomplex for M (A ) were given in [31] and in [13]. We review the material of [13, §4] that will be useful for our later purposes.…”