Centering toric arrangements of maximal rank

TL;DR

This paper proves that the complement manifolds of maximal rank toric arrangements are diffeomorphic to centered arrangements, establishing their minimality using linear algebra techniques.

Contribution

It introduces a matrix-based approach to classify maximal rank toric arrangements and proves their complement manifolds are topologically minimal and diffeomorphic to centered arrangements.

Findings

Complement manifolds of maximal rank arrangements are minimal.

Diffeomorphism to centered arrangements established.

Linear algebra methods used for classification.

Abstract

The homotopy type of the complement manifold of a complexified toric arrangement has been investigated by d'Antonio and Delucchi in a paper that shows the minimality of such topological space. In this work we associate to a given toric arrangement a matrix that represents the arrangement over the integers. Then, we consider the family of toric arrangements for which this matrix has maximal rank. Our goal is to prove, by means of basic linear algebra arguments, that the complement manifold of the toric arrangements that belong to this family is diffeomorphic to that of centered toric arrangements and thus it is a minimal topological space, too.