{
  "id": "2407.12156",
  "version": "v1",
  "published": "2024-07-16T20:24:30.000Z",
  "updated": "2024-07-16T20:24:30.000Z",
  "title": "Discrete Morse theory on $ΩS^2$",
  "authors": [
    "Lacey Johnson",
    "Kevin Knudson"
  ],
  "comment": "15 pages, 1 figure",
  "categories": [
    "math.AT"
  ],
  "abstract": "A classical result in Morse theory is the determination of the homotopy type of the loop space of a manifold. In this paper, we study this result through the lens of discrete Morse theory. This requires a suitable simplicial model for the loop space. Here, we use Milnor's $\\textrm{F}^+\\textrm{K}$ construction to model the loop space of the sphere $S^2$, describe a discrete gradient on it, and identify a collection of critical cells. We also compute the action of the boundary operator in the Morse complex on these critical cells, showing that they are potential homology generators. A careful analysis allows us to recover the calculation of the first homology of $\\Omega S^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-16T20:24:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q70",
      "55P35"
    ],
    "keywords": [
      "discrete morse theory",
      "loop space",
      "critical cells",
      "potential homology generators",
      "morse complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}