{
  "id": "1903.01753",
  "version": "v1",
  "published": "2019-03-05T09:58:51.000Z",
  "updated": "2019-03-05T09:58:51.000Z",
  "title": "Deformations of smooth functions on $2$-torus",
  "authors": [
    "Bohdan Feshchenko"
  ],
  "comment": "15 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $f $ be a Morse function on a smooth compact surface $M$ and $\\mathcal{S}'(f)$ be a group of $f$-preserving diffeomorphisms of $M$ which are isotopic to the identity map. Let also $G(f)$ be a group of automorphisms of the graph of $f$ induced by elements from $\\mathcal{S}'(f)$, and $\\Delta'$ be a subgroup of $\\mathcal{S}'(f)$ of diffeomorphisms which trivially act on the graph of $f$ and are isotopic to the identity map. The group $\\pi_0\\mathcal{S}'(f)$ can be viewed as an analogue of a mapping class group for $f$-preserved diffeomorphisms of $M$. Groups $\\pi_0\\Delta'(f)$ and $G(f)$ can be viewed as groups which encode `combinatorially trivial' and `combinatorially nontrivial' counterparts of $\\pi_0\\mathcal{S}'(f)$ respectively. In the paper we compute groups $\\pi_0\\mathcal{S}'(f)$, $G(f)$, and $\\pi_0\\Delta'(f)$ for Morse functions on $2$-torus $T^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-05T09:58:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S05",
      "20E22",
      "58B05"
    ],
    "keywords": [
      "smooth functions",
      "deformations",
      "identity map",
      "morse function",
      "diffeomorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}