{
  "id": "2402.12116",
  "version": "v1",
  "published": "2024-02-19T13:11:10.000Z",
  "updated": "2024-02-19T13:11:10.000Z",
  "title": "Discrete Morse theory for open complexes",
  "authors": [
    "Kevin P. Knudson",
    "Nicholas A. Scoville"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "We develop a discrete Morse theory for open simplicial complexes $K=X\\setminus T$ where $X$ is a simplicial complex and $T$ a subcomplex of $X$. A discrete Morse function $f$ on $K$ gives rise to a discrete Morse function on the order complex $S_K$ of $K$, and the topology change determined by $f$ on $K$ can be understood by analyzing the the topology change determined by the discrete Morse function on $S_K$. This topology change is given by a structure theorem on the level subcomplexes of $S_K$. Finally, we show that the Borel-Moore homology of $K$, a homology theory for locally compact spaces, is isomorphic to the homology induced by a gradient vector field on $K$ and deduce corresponding weak Morse inequalities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-19T13:11:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q70",
      "55U10",
      "55N35"
    ],
    "keywords": [
      "discrete morse theory",
      "discrete morse function",
      "open complexes",
      "topology change",
      "deduce corresponding weak morse inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}