Buchstaber invariants of skeleta of a simplex

Abstract

A moment-angle complex $\mathcal{Z}_{K}$ is a compact topological space associated with a finite simplicial complex $K$. It is realized as a subspace of a polydisk $(D^{2})^{m}$, where $m$ is the number of vertices in $K$ and $D^{2}$ is the unit disk of the complex numbers $\mathbb{C}$, and the natural action of a torus $(S^{1})^{m}$ on $(D^{2})^{m}$ leaves $\mathcal{Z}_{K}$ invariant. The Buchstaber invariant $s(K)$ of $K$ is the largest integer for which there is a subtorus of rank $s(K)$ acting on $\mathcal{Z}_{K}$ freely. The story above goes over the real numbers $\mathbb{R}$ in place of $\mathbb{C}$ and a real analogue of the Buchstaber invariant, denoted $s_{\mathbb{R}}(K)$, can be defined for $K$ and $s(K)\leqq s_{\mathbb{R}}(K)$. In this paper we will make some computations of $s_{\mathbb{R}}(K)$ when $K$ is a skeleton of a simplex. We take two approaches to find $s_{\mathbb{R}}(K)$ and the latter one turns out to be a problem of integer linear programming and of independent interest.