{
  "id": "1708.01778",
  "version": "v1",
  "published": "2017-08-05T15:20:03.000Z",
  "updated": "2017-08-05T15:20:03.000Z",
  "title": "The strong ring of simplicial complexes",
  "authors": [
    "Oliver Knill"
  ],
  "comment": "40 pages, 8 figures",
  "categories": [
    "math.CO",
    "cs.DM",
    "math.AT"
  ],
  "abstract": "We define a ring R of geometric objects G generated by finite abstract simplicial complexes. To every G belongs Hodge Laplacian H as the square of the Dirac operator determining its cohomology and a unimodular connection matrix L). The sum of the matrix entries of the inverse of L is the Euler characteristic. The spectra of H as well as inductive dimension add under multiplication while the spectra of L multiply. The nullity of the Hodge of H are the Betti numbers which can now be signed. The map assigning to G its Poincare polynomial is a ring homomorphism from R the polynomials. Especially the Euler characteristic is a ring homomorphism. Also Wu characteristic produces a ring homomorphism. The Kuenneth correspondence between cohomology groups is explicit as a basis for the product can be obtained from a basis of the factors. The product in R produces the strong product for the connection graphs and leads to tensor products of connection Laplacians. The strong ring R is also a subring of the full Stanley-Reisner ring S Every element G can be visualized by its Barycentric refinement graph G1 and its connection graph G'. Gauss-Bonnet, Poincare-Hopf or the Brouwer-Lefschetz extend to the strong ring. The isomorphism of R with a subring of the strong Sabidussi ring shows that the multiplicative primes in R are the simplicial complexes and that every connected element in the strong ring has a unique prime factorization. The Sabidussi ring is dual to the Zykov ring, in which the Zykov join is the addition. The connection Laplacian of the d-dimensional lattice remains invertible in the infinite volume limit: there is a mass gap in any dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-05T15:20:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C99",
      "13F55",
      "55U10",
      "68R05"
    ],
    "keywords": [
      "strong ring",
      "ring homomorphism",
      "euler characteristic",
      "finite abstract simplicial complexes",
      "barycentric refinement graph g1"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}