Topological Criteria for $k-$Formal Arrangements

TL;DR

This paper establishes a topological criterion for determining the $k$-formality of arrangements using a complex from vector spaces, with an application to graphic arrangements linked to graph homology.

Contribution

It introduces a new topological criterion for $k$-formality of arrangements and simplifies the characterization for graphic arrangements via graph homology.

Findings

$k$-formality of graphic arrangements is characterized by vanishing homology groups.

Provides a topological criterion linking arrangement formality to graph complex homology.

Simplifies the understanding of $k$-formality in terms of combinatorial and topological properties.

Abstract

We prove a criterion for $k-$formality of arrangements, using a complex constructed from vector spaces introduced in \cite{bt}. As an application, we give a simple description of $k-$formality of graphic arrangements: Let $G$ be a connected graph with no loops or multiple edges. Let $\Delta$ be the flag (clique) complex of $G$ and let $H_{\bullet}(\Delta)$ be the homology of the chain complex of $\Delta$. If $\mathcal A_G$ is the graphic arrangement associated to $G$, we will show that $\mathcal A_G$ is $k-$formal if and only if $H_i(\Delta)=0$ for every $i=1,...,k-1$.