{
  "id": "1911.04208",
  "version": "v1",
  "published": "2019-11-11T12:20:10.000Z",
  "updated": "2019-11-11T12:20:10.000Z",
  "title": "Poincare Hopf for vector fields on graphs",
  "authors": [
    "Oliver Knill"
  ],
  "comment": "15 pages, 5 figures",
  "categories": [
    "math.CO",
    "cs.DM",
    "math.DS"
  ],
  "abstract": "We generalize the Poincare-Hopf theorem sum_v i(v) = X(G) to vector fields on a finite simple graph (V,E) with Whitney complex G. To do so, we define a directed simplicial complex as a finite abstract simplicial complex equipped with a bundle map F: G to V telling which vertex T(x) in x dominates the simplex x. The index i(v) of a vertex v is defined as X(F^-1(v)), where X is the Euler characteristic. We get a flow by adding a section map F: V to G. The resulting map T on G is a discrete model for a differential equation x'=F(x) on a compact manifold. Examples of directed complexes are defined by Whitney complexes defined by digraphs with no cyclic triangles or gradient fields on finite simple graphs defined by a locally injective function. The result extends to simplicial complexes equipped with an energy function H:G to Z that implements a divisor. The index sum is then the total energy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-11T12:20:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "57M15",
      "03H05",
      "62-07",
      "62-04"
    ],
    "keywords": [
      "vector fields",
      "poincare hopf",
      "finite simple graph",
      "finite abstract simplicial complex",
      "whitney complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}