{
  "id": "1111.0506",
  "version": "v1",
  "published": "2011-11-02T14:03:04.000Z",
  "updated": "2011-11-02T14:03:04.000Z",
  "title": "Extensions of Cantor minimal systems and dimension groups",
  "authors": [
    "Eli Glasner",
    "Bernard Host"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Given a factor map $p : (X,T) \\to (Y,S)$ of Cantor minimal systems, we study the relations between the dimension groups of the two systems. First, we interpret the torsion subgroup of the quotient of the dimension groups $K_0(X)/K_0(Y)$ in terms of intermediate extensions which are extensions of $(Y,S)$ by a compact abelian group. Then we show that, by contrast, the existence of an intermediate non-abelian finite group extension can produce a situation where the dimension group of $(Y,S)$ embeds into a proper subgroup of the dimension group of $(X,T)$, yet the quotient of the dimension groups is nonetheless torsion free. Next we define higher order cohomology groups $H^n(X \\mid Y)$ associated to an extension, and study them in various cases (proximal extensions, extensions by, not necessarily abelian, finite groups, etc.). Our main result here is that all the cohomology groups $H^n(X \\mid Y)$ are torsion groups. As a consequence we can now identify $H^0(X \\mid Y)$ as the torsion group of $ K_0(X)/K_0(Y)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-02T14:03:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dimension group",
      "cantor minimal systems",
      "intermediate non-abelian finite group extension",
      "define higher order cohomology groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.0506G"
    }
  }
}