{
  "id": "1711.09527",
  "version": "v1",
  "published": "2017-11-27T04:25:44.000Z",
  "updated": "2017-11-27T04:25:44.000Z",
  "title": "One can hear the Euler characteristic of a simplicial complex",
  "authors": [
    "Oliver Knill"
  ],
  "comment": "9 pages, 2 figures",
  "categories": [
    "math.CO",
    "cs.DM",
    "math.GN"
  ],
  "abstract": "We prove that that the number p of positive eigenvalues of the connection Laplacian L of a finite abstract simplicial complex G matches the number b of even dimensional simplices in G and that the number n of negative eigenvalues matches the number f of odd-dimensional simplices in G. The Euler characteristic X(G) of G therefore can be spectrally described as X(G)=p-n. This is in contrast to the more classical Hodge Laplacian H which acts on the same Hilbert space, where X(G) is not yet known to be accessible from the spectrum of H. Given an ordering of G coming from a build-up as a CW complex, every simplex x in G is now associated to a unique eigenvector of L and the correspondence is computable. The Euler characteristic is now not only the potential energy summing over all g(x,y) with g=L^{-1} but also agrees with a logarithmic energy tr(log(i L)) 2/(i pi) of the spectrum of L. We also give here examples of L-isospectral but non-isomorphic abstract finite simplicial complexes. One example shows that we can not hear the cohomology of the complex.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-27T04:25:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C99",
      "55U10",
      "68R05"
    ],
    "keywords": [
      "euler characteristic",
      "non-isomorphic abstract finite simplicial complexes",
      "finite abstract simplicial complex",
      "logarithmic energy tr",
      "classical hodge laplacian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}