Topological Criteria for $k-$Formal Arrangements

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

9 pages, 1 figure