On the homotopy type of the complement of an arrangement that is a 2-generic section of the parallel connection of an arrangement and a pencil of lines

TL;DR

This paper investigates the homotopy type of complements of certain line arrangements in complex space, showing that while their complements may not be diffeomorphic, they are always homotopy equivalent.

Contribution

It establishes that complements of arrangements formed by a 2-generic section of a parallel connection are homotopy equivalent despite not being diffeomorphic.

Findings

Homotopy equivalence of arrangement complements is guaranteed.

Differences in diffeomorphism types do not affect homotopy type.

Arrangement construction involves a pencil of lines with specific intersection properties.

Abstract

Let $\mathcal{A} $ be a complexified-real arrangement of lines in $\mathbb{C}^2.$ Let $H$ be any line in $ \mathcal{A} $. Then, form a new complexified-real arrangement $ \mathcal{B}_H = \mathcal{A} \cup \mathcal{C} $ where $ \mathcal{C} \cup \{H\} $ is a pencil of lines with multiplicity $ m\geq 3 $, the intersection point in the pencil is not a multiple point in $ \mathcal{A}, $ and every line in $ \mathcal{C} $ intersects every line in $ \mathcal{A}\setminus \{H\} $ in points of multiplicity two. In this article, we show that for $ H_1, H_2 \in \mathcal{A} $ we may have that $ \mathcal{B}_{H_1} $ and $ \mathcal{B}_{H_2} $ do not have diffeomorphic complements, but the complements of the arrangements will always be homotopy equivalent.