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

Kristopher Williams

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arXiv · 1507.04706

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

Kristopher Williams

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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.

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Kristopher Williams. 2015-07-16. 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. https://arxiv.org/abs/1507.04706

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