Topological Criteria for $k-$Formal Arrangements

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 $i=1,...,k-1$.