Minimality of toric arrangements

Abstract

We prove that the complement of a toric arrangement has the homotopy type of a minimal CW complex. As a corollary we obtain that the integer cohomology of these spaces is torsion free.We use Discrete Morse Theory, providing a sequence of cellular collapses that leads to a minimal complex.

The integer cohomology algebra of toric arrangements

“…Structure of the paper. Given a complexified toric arrangement A (defined in §2.1), our combinatorial model for the homotopy type of the complement of A is the toric Salvetti complex SalpAq, in the formulation given in [9], in particular as the nerve of an acyclic category obtained as homotopy colimit of a diagram of posets. In Section 3 we review some basic facts about the combinatorics and topology of acyclic categories and establish some facts about the combinatorial topology of Salvetti complexes of complexified hyperplane arrangements.…”

“…We refer e.g. to [9,Section 3] for a precise definition and here only recall that the face category FpKq of a polyhedral complex K has the cells of K as objects, and one morphism P Ñ Q for every attachment of the polyhedral cell P to a face of the polyhedral cell Q.…”

“…In fact, much of the notation of the previous section has been introduced in [10,9] in order to describe the relationships involved in these covering morphisms. For instance, if F P FpAq, the arrangement ArF s is an 'abstract' copy of every A ae for which we can choose linear forms according to those defining A ae .…”

“…The order relation F ae ď G ae also defines an inclusion jrF ae ď G ae s : SalpA ae G ae q ãÑ SalpA ae F ae q, i.e., the map induced on complexes by the inclusion j G of Definition 3.3.2 with respect to the "ambient" arrangement A We can now re-state the following definition from [9]. …”

“…A first combinatorial model for the homotopy type of complements of toric arrangements was introduced by Moci and Settepanella [26] for "centered" arrangements (i.e., defined by kernels of characters) which induce a regular CW-decomposition of the compact torus pS 1 q d Ď pC˚q d , and was subsequently generalized to the case of "complexified" toric arrangements (S 1 -level sets of characters) by d'Antonio and the second author [10] who, on this basis, also gave a presentation of the complement's fundamental group. In later work [9], d'Antonio and the second author also proved that complements of complexified toric arrangements are minimal spaces (i.e., they have the homotopy of a CW-complex where the i-dimensional cells are counted by the i-th Betti number): in particular, the integer cohomology groups are torsion-free and are thus determined by the associated arithmetic matroid. This raises the question of whether, as is the case with the OS-Algebra of hyperplane arrangements, the integer cohomology ring is combinatorially determined.…”