Combinatorics of Toric Arrangements

TL;DR

This paper develops an algebraic model for the cohomology of toric arrangement complements, explores the dependence on combinatorial structures, and identifies conditions for topological equivalences and algebraic generation.

Contribution

It introduces an Orlik-Solomon model for toric arrangements and analyzes how it depends on the arrangement's combinatorial data and poset of layers.

Findings

The cohomology algebra can be generated in degree one under certain conditions.

Conditions are identified under which two arrangements have diffeomorphic complements.

The model's dependence on the arrangement's combinatorial and poset structure is clarified.

Abstract

In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of arithmetic matroids, in order to discuss how the Orlik-Solomon model depends on the poset of layers. The analysis of discriminantal toric arrangements permits us to isolate certain conditions under which two toric arrangements have diffeomorphic complements. We also give combinatorial conditions determining whether the cohomology algebra is generated in degree one.