Two Examples of Toric Arrangements

TL;DR

This paper demonstrates that the integral cohomology algebra of toric arrangement complements cannot be fully characterized by the poset of layers, and the rational cohomology algebra is not determined by the arithmetic matroid.

Contribution

It provides counterexamples showing the limitations of combinatorial invariants in determining cohomology algebras of toric arrangements.

Findings

Integral cohomology algebra is not determined by the poset of layers.

Rational cohomology algebra is not determined by the arithmetic matroid.

Poset of layers determines rational cohomology algebra.

Abstract

We show that the integral cohomology algebra of the complement of a toric arrangement is not determined by the poset of layers. Moreover, the rational cohomology algebra is not determined by the arithmetic matroid (however it is determined by the poset of layers).