On the connection between coordinate and diagonal arrangement complements

TL;DR

This paper explores the topological relationship between coordinate and diagonal arrangement complements, establishing suspension relations for specific classes of simplicial complexes in complex and real spaces.

Contribution

It proves that for certain simplicial complexes, the coordinate arrangement complement is a double suspension of the diagonal arrangement complement in complex spaces.

Findings

U(K) is the double suspension of D(K) in complex spaces.

U_R(K) is the single suspension of D_R(K) in real spaces.

The class of complexes considered has missing faces sharing a common vertex.

Abstract

We study diagonal arrangement complements $D(K)$ in $\mathbb{C}^m$. We consider the class of simplicial complexes $K$ in which any two missing faces have a common vertex, and prove that the coordinate arrangement complement $U(K)$ is the double suspension of the diagonal arrangement complement $D(K)$. In the case of subspace arrangements in $\mathbb{R}^m$ the coordinate arrangement complement $U_{\mathbb{R}}(K)$ is the single suspension of $D_{\mathbb{R}}(K)$.