Topological criteria for \(\mathbf k\)-formal arrangements (Q880220)

Let \(\{\alpha_1,\dots,\alpha_n\}\) be a set of linear forms on a vector space \(V,\) defining a hyperplane arrangement \(\mathcal A.\) Let \(F({\mathcal A})\) be the space of linear relations among the \(\alpha_i,\) and \(R_2({\mathcal A})\subseteq F({\mathcal A})\) the subspace spanned by relations involving at most three \(\alpha_i.\) Equivalently, \(R_2({\mathcal A})=\bigoplus_{r(X)=2} F({\mathcal A}_X)\) where \({\mathcal A}_X\) is the collection of \(\alpha_i\) whose kernels contain \(X,\) and \(r(X)\) is the codimension of \(X\) in \(V.\) The arrangement \(\mathcal A\) is formal (now 2-formal) if \(F({\mathcal A})=R_2({\mathcal A}).\) This notion was generalized by \textit{K.~Brandt} and \textit{H.~Terao} in [Discrete Comput. Geom. 12, 49--63 (1994; Zbl 0802.52010)]. The higher relation spaces \(R_k({\mathcal A}), k\geq 3,\) are defined inductively: \(R_k({\mathcal A})\) is the kernel of \(\bigoplus_{r(X)=k-1} R_{k-1}({\mathcal A}_X) \to R_{k-1}({\mathcal A}).\) The arrangement \(\mathcal A\) is \(k\)-formal if \(\mathcal A\) is \((k-1)\)-formal and \(\bigoplus_{r(X)=k} R_k({\mathcal A}_X) \to R_k({\mathcal A})\) is surjective. For instance, \({\mathcal A}\) is 3-formal if all relations among the \(\alpha_i\) are consequences of those supported locally, on rank-two flats, and all relations among these relations are consequences of those supported locally, on rank-three flats. Brandt and Terao proved: if \({\mathcal A}\) is a free arrangement, then \({\mathcal A}\) is \(k\)-formal for all \(k.\) In the paper under review, the author observes that these relation spaces and inclusions can be put together into a chain complex, whose homology vanishes through degree \(k\) if and only if \(\mathcal A\) is \(k\)-formal. Then, if \({\mathcal A}={\mathcal A}_G\) is the graphic arrangement determined by a graph \(G,\) it is shown that the chain complex of relations is isomorphic to the complex of simplicial chains on the clique (or flag) complex \(\Delta(G)\) of \(G.\) Thus \({\mathcal A}_G\) is \(k\)-formal if and only if \(\Delta(G)\) is homologically \(k\)-connected. Reviewer's comments: The main result of the paper gives an alternate proof that the flag complex of a chordal graph is acyclic. Another corollary is that the graphic arrangement \({\mathcal A}_G\) is \(k\)-formal if and only if the Bestvina-Brady group associated with \(G\) is of type \(FP_{k+1}\) over the integers. This may or may not be merely coincidental.