Criterion for the Buchstaber invariant of simplicial complexes to be equal to two

Abstract

In this paper we study the Buchstaber invariant of simplicial complexes, which comes from toric topology. With each simplicial complex $K$ on $m$ vertices we can associate a moment-angle complex ${\mathcal{Z}}_K$ with a canonical action of the compact torus $T^m$. Then $s(K)$ is the maximal dimension of a toric subgroup that acts freely on ${\mathcal{Z}}_K$. We develop the Buchstaber invariant theory from the viewpoint of the set of minimal non-simplices of $K$. It is easy to show that $s(K)=1$ if and only if any two and any three minimal non-simplices intersect. For $K=\partial P^{\ast }$, where $P$ is a simple polytope, this implies that $P$ is a simplex. The case $s(P)=2$ is such more complicated. For example, for any $k⩾2$ there exists an $n$-polytope with $n+k$ facets such that $s(P)=2$. Our main result is the criterion for the Buchstaber invariant of a simplicial complex $K$ to be equal to two.

8 pages

References in corpus (1)

• The problem of Buchstaber number and its combinatorial aspects