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 $\C$, 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 maximum 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 $\C$ 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.

Cited by in corpus (3)

• Moment-angle complexes from simplicial posets
• Moment-angle complexes and polyhedral products for convex polytopes
• Simplicial complexes Alexander dual to boundaries of polytopes