{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a compact connected polyhedron, that is, a space homeomorphic to a finite simplicial complex.\nLet $f \\colon X \\to X$ be a [[continuous map|0VGFHTM75Y]] with fixed point set $\\mathrm{Fix}(f) = \\set{x \\in X : f(x) = x}$.\n\n1. Two fixed points $x, y$ are *Nielsen equivalent* if there is a path $c$ from $x$ to $y$ such that $f \\circ c$ is [[path homotopic|CCNKA8C711]] to $c$.\n2. The equivalence classes are the *fixed point classes* of $f$; there are finitely many, and each is open and closed in $\\mathrm{Fix}(f)$.\n3. Each fixed point class $F$ has a *fixed-point index* $i(f, F) \\in \\Z$.\n   For a single fixed point $x$ in an open set of $\\R^n$, it is the local degree of $u \\mapsto u - f(u)$ at $x$.\n4. A class $F$ is *essential* if $i(f, F) \\neq 0$.\n\nThe *Nielsen number* $N(f)$ is the number of essential fixed point classes of $f$.\n\n## Remarks\n\nThe sum of the indices of all fixed point classes is the [[Lefschetz number|ZSYCA4EGEZ]] $L(f)$.\nThe number $N(f)$ depends only on the homotopy class of $f$, and every map [[homotopic|F2F143KQ2M]] to $f$ has at least $N(f)$ fixed points.\n\nExample: on the circle, let $f(z) = z^k$ with $k \\neq 1$.\nThe fixed points are the $\\abs{k - 1}$ roots of $z^{k-1} = 1$; they lie in distinct classes, each of index $\\pm 1$, so $N(f) = \\abs{1 - k} = \\abs{L(f)}$.\n\nNon-example of essentiality: for the identity of $S^1$, the single fixed point class is all of $S^1$.\nIts index is $L(\\id) = \\chi(S^1) = 0$, so it is not essential.\nIndeed a small rotation has no fixed point, and $N(\\id) = 0$.\n\nFor compact manifolds of dimension at least $3$, Wecken's theorem says that some map homotopic to $f$ has exactly $N(f)$ fixed points.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "GHXMVWN3W5",
    "kind": "definition",
    "msc": [
      "55M20",
      "54H25"
    ],
    "name": "Nielsen fixed-point theory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fixed-point_theorem"
      }
    ],
    "tag": "GEQV0DZS3W",
    "uses": [
      "0VGFHTM75Y",
      "CCNKA8C711",
      "F2F143KQ2M",
      "ZSYCA4EGEZ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/GEQV0DZS3W/",
  "links": [
    {
      "field": "defines",
      "target": "GHXMVWN3W5"
    },
    {
      "field": "uses",
      "target": "0VGFHTM75Y"
    },
    {
      "field": "uses",
      "target": "CCNKA8C711"
    },
    {
      "field": "uses",
      "target": "F2F143KQ2M"
    },
    {
      "field": "uses",
      "target": "ZSYCA4EGEZ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GEQV0DZS3W.md",
  "msc": [
    "55M20",
    "54H25"
  ],
  "name": "Nielsen fixed-point theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GEQ/GEQV0DZS3W.md",
  "status": null,
  "tag": "GEQV0DZS3W",
  "uri": "m2://GEQV0DZS3W",
  "url": "https://math2.tamnd.com/wiki/nielsen-fixed-point-theory-definition/",
  "writing": {
    "tag": "GEQV0DZS3W",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}