Discrete topological methods for subspace arrangements (Q2906436)

Let \(\mathcal{A}=\{H_1, H_2,\dots, H_r\}\) be a central hyperplane arrangement in \(\mathbb{R}^n\). Write NEWLINENEWLINE\[NEWLINEH_i=\big\{x\in M\;\big|\; a_j\cdot x=0\big\},NEWLINE\]NEWLINENEWLINENEWLINE where \(a_j\in\mathbb{R}^n\setminus\{0\}\). For each integer \(d>0\), one can define the \(d\)-complexification \(\mathcal{A}^{(d)}\) of \(\mathcal{A}\) as the finite collection of codimension-\(d\) linear subspaces \(H_j^{(d)}\) of \((\mathbb{R}^n)^d\)NEWLINENEWLINENEWLINE\[NEWLINEH_j^{(d)}=\big\{(x_1, \dots, x_d)\;\big|\; a_j\cdot x_k=0,\quad k=1, \dots, d\big\}.NEWLINE\]NEWLINENEWLINENEWLINE It generalizes the standard complexification of real arrangements which is the same as the \(2\)-complexification. The complement NEWLINE\[NEWLINE\mathcal{M}(\mathcal{A})^{(d)}=(\mathbb{R}^n)^d\setminus \bigcup\limits_{H\in\mathcal{A}}H^{(d)}NEWLINE\]NEWLINENEWLINENEWLINE is called the generalized configuration space associated to \(\mathcal{A}\).NEWLINENEWLINEIt is known that \(\mathcal{M}(\mathcal{A})^{(2)}\) is homotopic to a minimal CW-complex, i.e., the number of \(i\)-cells is equal to the \(i\)-th Betti number for each \(i\). Applying discrete Morse theory, in [Geom. Topol. 11, 1733--1766 (2007; Zbl 1134.32010)] \textit{M. Salvetti} and \textit{S. Settepanella} give an explicit description of the minimal CW-complex structure of \(\mathcal{M}(\mathcal{A})^{(2)}\). The authors of the paper under review extend the techniques for \(d=2\) to \(d>2\) and show that the configuration space \(\mathcal{M}(\mathcal{A})^{(d)}\) for each \(d\) admits an explicit minimal CW-complex. Detailed examples are presented while a detailed proof will appear elsewhere due to the technicality.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14003].