{
  "id": "1212.3970",
  "version": "v1",
  "published": "2012-12-17T12:24:56.000Z",
  "updated": "2012-12-17T12:24:56.000Z",
  "title": "Criterion for the Buchstaber invariant of simplicial complexes to be equal to two",
  "authors": [
    "Nickolai Erokhovets"
  ],
  "comment": "8 pages",
  "categories": [
    "math.AT",
    "math.CO"
  ],
  "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 $K=\\partial P^*$, 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 $k\\geqslant 2$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-17T12:24:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Nxx",
      "52Bxx"
    ],
    "keywords": [
      "simplicial complexes",
      "minimal non-simplices intersect",
      "buchstaber invariant theory",
      "maximal dimension",
      "main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.3970E"
    }
  }
}