On torsion freeness for the decomposable Orlik-Solomon algebra

TL;DR

This paper proves that the decomposable Orlik--Solomon algebra of a simple matroid is torsion free, which has implications for understanding the topology of complex hyperplane arrangement complements.

Contribution

It establishes the torsion freeness of the decomposable Orlik--Solomon algebra for simple matroids, linking combinatorial properties to topological invariants.

Findings

Proves torsion freeness of the algebra for simple matroids.

Shows impact on higher homotopy groups of arrangement complements.

Connects algebraic properties to topological features of hyperplane arrangements.

Abstract

We prove the torsion freeness of the decomposable Orlik--Solomon algebra of a simple matroid on ground set $[n]$. In the class of hypersolvable \& non-supersolvable complex hyperplane arrangements, the torsion freeness, in a certain degree, of this combinatorially defined object, associated to the intersection lattice of the arrangement, impacts on the first non-vanishing higher homotopy group of the complement of the arrangement.