Restrictions of aspherical arrangements

TL;DR

This paper provides examples of hyperplane arrangements where the property of asphericity is not preserved under restrictions, demonstrating that asphericity is not a hereditary property.

Contribution

It introduces specific examples of $K(Cpi,1)$-arrangements with restrictions that are not $K(Cpi,1)$, highlighting a new limitation in the theory of hyperplane arrangements.

Findings

Asphericity is not hereditary among hyperplane arrangements.

Certain restrictions of $K(Cpi,1)$-arrangements fail to be $K(Cpi,1)$.

The paper provides explicit counterexamples.

Abstract

In this note we present examples of $K(\pi,1)$-arrangements which admit a restriction which fails to be $K(\pi,1)$. This shows that asphericity is not hereditary among hyperplane arrangements.