{
  "id": "math/0109073",
  "version": "v1",
  "published": "2001-09-11T20:42:59.000Z",
  "updated": "2001-09-11T20:42:59.000Z",
  "title": "Augmental Homology and the Kynneth Formula for Joins",
  "authors": [
    "G. Fors"
  ],
  "comment": "These 32 pages has been prepared using AMSTeX with \\documentstyle{amsppt} in a MikTeX2.0 environment. It's an improvement based on REPORTS/ Department of Mathematics, University of Stockholm, Sweden; A Homology Theory Based on the Existence of a (-1)-dimensional Simplex; by G. Fors, 1994 - No 3. (25 pages)",
  "categories": [
    "math.AT",
    "math.CO"
  ],
  "abstract": "The \"simplicial complexes\" and \"join\" (*) today used within combinatorics aren't the classical concepts, cf. Spanier (1966) p. 108-9, but, exept for \\emptyset, complexes having {\\emptyset} as a subcomplex resp. \\Sigma1 * \\Sigma2 := {\\sigma1 \\cup \\sigma2 | \\sigmai \\in \\Sigmai} implying a tacit change of unit element w.r.t. the join operation, from \\emptyset to {\\emptyset}. Extending the classical realization functor to this category of simplicial complexes we end up with a \"restricted\" category of topological spaces, \"containing\" the classical and where the classical (co)homology theory, as well as the ad-hoc invented reduced versions, automatically becomes obsolete, in favor of a unifying and more algebraically efficient theory. This very modest category modification greatly improves the interaction between algebra and topology. E.g. it makes it possible to calculate the homology groups of a topological pair-join, expressed in the relative factor groups, leading up to a truly simple boundary formula for joins of manifolds: Bd(X1 * X2) = ((BdX1 * X2) \\cup (X1 * BdX2)), the product counterpart of which is true also classically. It is also easily seen that no finite simplicial n-manifold has an (n-2)-dimensional boundary, cf. Cor. 1 p. 26, and that simplicial homology manifolds with the integers as koefficient module are all locally orientable, cf. Cor. 2 p. 29.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-09-11T20:42:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Nxx",
      "55N10",
      "57P05"
    ],
    "keywords": [
      "augmental homology",
      "kynneth formula",
      "simplicial complexes",
      "simplicial homology manifolds",
      "truly simple boundary formula"
    ],
    "note": {
      "typesetting": "AMS-TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......9073F"
    }
  }
}