On the Falk invariant of signed graphic arrangements (Q1637101)

In this interesting paper, the authors find a combinatorial description for the 3rd Falk invariant of signed graphic arrangements that do not have a \(B_{2}\) sub-arrangement. Let \(\mathcal{A} = \{H_{1},\dots,H_{n}\} \subset \mathbb{C}^{\ell}\) be a hyperplane arrangement and we denote by \(E^{1} = \bigoplus_{j=1}^{n} \mathbb{C}e_{j}\) the free module generated by \(e_{1},e_{2},\dots,e_{n}\), where \(e_{i}\) is a symbol corresponding to \(H_{i}\). Let \(E = \bigwedge E^{1}\) be the exterior algebra over \(\mathbb{C}\). Denote by \(I(A)\) the Orlik-Solomon ideal associated to \(\mathcal{A}\) and we define the Orlik-Solomon algebra of \(\mathcal{A}\) by \(A:= E / I(\mathcal{A})\). We can write \(A = \bigoplus_{p\geq 0} A^{p}\), where \(A^{p} = E^{p} / I^{p}\) and \(I^{p} = I\cap E^{p}\) due to the homogenity of \(I\). We are ready to define the (3rd) Falk invariant. Definition 1. Consider the map \(d\) defined by \[ d: E^{1} \otimes I^{2} \rightarrow E^{3} \] \[ d(a\otimes b) = a \wedge b. \] The Falk invariant is defined as \(\phi_{3} := \dim(\text{ker} (d))\). It is known that \(\phi_{3}\) can be described from the lower central series of the fundamental group \(\pi(M)\) of the complex complement \(M\) of the arrangement \(\mathcal{A}\). If we consider the lower central series as a chain of subgroups \(N_{i}\), where \(N_{1} = \pi(M)\), and \(N_{k+1} = [N_{k},N_{1}]\), the subgroup generated by commutators of elements in \(N_{k}\) and \(N_{1}\), then \(\phi_{3}\) is the rank of the finitely generated abelian group \(N_{3} / N_{4}\). Now we look at the signed graphic arrangements. Definiton 2. A signed graph is a tuple \(G = (V_{G}, E_{G}^{+}, E_{G}^{-},L_{G})\), where \(V_{G}\) is a finite set called the set of vertices, \(E_{G}^{+}\) is a subset of \({V_{g} \choose 2}\) called the set of positive edges, \(E_{G}^{-}\) is a subset of \({V_{G} \choose 2}\) called the set of negative edges, and \(L_{G}\) is a subset of \(V_{G}\) called the set of loops. Definition 3. Given a signed graph \(G\), let \(\mathcal{A}(G)\) be the hyperplane arrangement in \(\mathbb{C}^{\ell}\), where \(\ell\) is the number of vertices of \(G\), consisting of the following hyperplanes: {\parindent=8mm \begin{itemize}\item[(i)] \(\{x_{i}-x_{j} = 0\}\) for \(\{i,j\} \in E_{G}^{+}\), \item[(ii)] \(\{x_{i}+x_{j}=0 \}\) for \(\{i,j\} \in E_{G}^{-}\), \item[(iii)] \(\{x_{i}=0\}\) for \(i \in L_{G}\). \end{itemize}} We call \(\mathcal{A}(G)\) the signed graphic arrangement associated to \(G\). For a given signed graph \(G\) we define the sign function on the set of edges by \(\text{sign}(e) ={+}\) if \(e \in E_{G}^{+}\), and \(\text{sign}(e) ={-}\) if \(e \in E_{G}^{-}\cup L_{G}\). Given a signed graph \(G\) and a function \(\sigma : V_{G} \rightarrow \{+,-\}\), we can define a new signed graph \(G'\) that has the same underlying graph as \(G\) but with a different sign function. In particular, if \(e = \{i,j\} \in E_{G}\), then \(\text{sing}_{G'}(e) = \sigma(i) \text{sing}_{G}(e) \sigma(j)\). We call \(G'\) the switching of \(G\), and \(\sigma\) a switching function. In order to formulate the main result of the paper, we need to introduce the following notion: \(K_{\ell}\) is a complete graph with \(\ell\) vertices and all edges being positive, \(D_{\ell}\) is a complete signed graph with \(\ell\) vertices and no loops, \(B_{\ell}\) is a signed complete graph with \(\ell\) vertices and a full set of loops, \(K_{\ell}^{\ell}\) is a complete graph with \(\ell\) vertices, all edges being positive, and a full set of loops, \(D_{\ell}^{1}\) is a signed complete graph with \(\ell\) vertices and one loop, and we define \(G_{\circ}\) as the graph having \(V_{G} = \{1,2,3\}\), \(L_{G} = \{2\}\), \(E_{G}^{+} = \{\{1,3\},\{1,2\},\{2,3\}\}\), \(E_{G}^{-} = \{\{1,2\},\{2,3\}\}\). Finally, we denote by \(\overline{G}\) a signed graph switching equivalent to \(G\) for some switching function \(\sigma\) (i.e., these are graphs having the same list of balanced cycles, where a cycle \(C = e_{1}e_{2}\dots e_{k}e_{1}\) is balanced if we have \(\text{sign}(C) :=\prod_{i=1}^{k}\text{sign}(e_{i}) ={+}\)). Main Theorem. For a signed graphic arrangement \(\mathcal{A}(G)\) associated to a signed graph \(G\) not containing a subgraph isomorphic to \(B_{2}\) subgraph, we have \[ \phi_{3} = 2(k_{3}+k_{4}+d_{3}+d_{2,1}+k_{2,2}+k_{3,3}+g_{\circ})+5d_{3,1}, \] where \(k_{l}\) denotes the number of subgraphs of \(G\) isomorphic to a \(\overline{K_{l}}\), \(d_{l}\) denotes the number of subgraphs of \(G\) isomorphic to \(D_{l}\) but not contained in \(D_{l}^{1}\), \(d_{l,1}\) denotes the number of subgraphs of \(G\) isomorphic to \(D_{l}^{1}\), \(k_{l,l}\) denotes the number of subgraphs of \(G\) isomorphic to \(\overline{K^{l}_{l}}\), and \(g_{\circ}\) denotes the number of subgraphs of \(G\) isomorphic to \(\overline{G_{\circ}}\) but not contained in \(D_{l}^{1}\).