Periodicity and the values of the real Buchstaber invariants
Consider a simplicial complex \(K\). The moment-angle space \(\mathcal{Z}_K\) is a special topological space defined in toric topology. It is a special example of a \(K\)-power. The natural action of \((S^1)^m\) on \(\mathbb{C}^m\) leaves the moment-angle complex \(\mathcal{Z}_k\) invariant and its action on \(\mathcal{Z}_k\) is, in general, non-free. The Buchstaber invariant \(s(K)\) is defined as the maximal dimension of a toric subgroup \(G\subset (S^1)^m\) which acts freely on \(\mathcal{Z}_k\). In a similar way, the real version \(s_{\mathbb R}(K)\) is defined by using the real moment-angle complex instead of the complex one. The invariants \(s(K)\) and \(s_{\mathbb R}(K)\) are useful in order to distinguish two simplicial complexes and they are a source of nontrivial and interesting combinatorial tasks. An important problem is finding a combinatorial description of \(s(K)\). Denote \(s_{\mathbb R}(m,p) = s_{\mathbb R}(\Delta^{m-1}_{m-p-1})\) where \(\Delta^{m-1}_{m-p-1}\) is the \((m-p-1)\)-skeleton of the \((m-1)\) simplex \(\Delta^{m-1}\). The problem of finding \(s_{\mathbb R}(m,p)\) can be transformed into a problem of integer programming. For a nonnegative integer \(b\geq 0\), one denotes by \(m_k(b)\) the maximum of \(\Sigma a_v\) over all nonnegative integers \(a_v, v\in (\mathbb{Z}/2)^k\backslash \{0\}\) satisfying \(\sum_{(u,v)=0}a_v\leq b\) for each \(u\in (\mathbb{Z}/2)^k\backslash \{0\}\). The problem of finding \(s_{\mathbb R}(m,p)\) is equivalent to that of finding \(m_k(b)\) for all \(k\). This problem can be formulated as an integer linear programming and \(m_k(b)\) can be written as a sum of optimal solutions. The main result is a formula involving \(m_k\). Theorem 1.1 Let \(k\geq 2\). Then \(m_k((2^{k-1}-1)\ell_2+b)-m_k((2^{k-1}-1)\ell_1+b)=(2^k-1)(\ell_2-\ell_1)\), for any nonnegative integers \(0\leq b\leq 2^{k-1}-2\), and \(\ell_1, \ell_2\) satisfying \(\ell_2\geq \ell_1\geq d_k(2^{k-1}-1-b)\). Next, in Theorem 1.2, an expansion formula for a certain term involved in Theorem 1.1 is obtained. After giving some preliminaries on \(m_k(b)\) and the problem of linear programming associated with \(m_k(b)\), the author proves the preperiodicity of \(m^*_k(b)\) and finds the formula for the values of \(m_k^*(b)\). Then the proofs of Theorems 1.1 and 1.2 are obtained. The author finds an upper bound of \(d_k(b)\) for each \(0\leq b\leq 2^{k-1}-1\). The paper concludes with some counterexamples and some concluding remarks.
moment-angle complex of a simplicial complex
scientific article; zbMATH DE number 6669094