Orlik-Solomon-type presentations for the cohomology algebra of toric arrangements

Abstract

We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes, and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont's rational formality result to formality over $\mathbb{Z}$. The data needed in order to state the presentation is fully encoded in the poset of connected components of intersections of the arrangement.

33 pages. In the statement of the main theorem the previous version only mentioned relations associated with the circuits in the matroid while we realized that subsets whose cardinality exceeds the rank by one may give new relations. Consequently some changes have been made in chapter 6

References in corpus (3)

• Cohomology of abelian arrangements
• Representation stability for the cohomology of arrangements associated to root systems
• Two Examples of Toric Arrangements

Cited by in corpus (12)

• $G$-Tutte polynomials and abelian Lie group arrangements
• Orientable arithmetic matroids
• Two Examples of Toric Arrangements
• On the cohomology of arrangements of subtori
• Homology of tropical fans
• Erratum to "The integer cohomology algebra of toric arrangements"
• The homotopy type of elliptic arrangements
• Inductive and divisional posets
• Finitary affine oriented matroids
• Cohomology ring of manifold arrangements
• The cohomology rings of the unordered configuration spaces of the torus
• Cohomology ring of non-compact abelian arrangements