{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f \\colon \\C^{n+1} \\to \\C$ be a nonconstant polynomial with $f(0) = 0$, where $n \\ge 0$.\nFor $\\varepsilon \u003e 0$ let $S_\\varepsilon$ be the [[sphere|P9CM4GQ2T0]] of radius $\\varepsilon$ about $0$ in $\\C^{n+1}$, and let $K = f^{-1}(0) \\cap S_\\varepsilon$.\n\n1. For all sufficiently small $\\varepsilon \u003e 0$, the map\n$$\\phi = \\frac{f}{\\abs{f}} \\colon S_\\varepsilon \\setminus K \\to S^1$$\nis a smooth locally trivial [[fibration|CX4M8K61AB]].\n2. Suppose moreover that $0$ is an isolated [[critical point|0YTCMFB4HC]] of $f$, with Milnor number\n$$\\mu = \\dim_\\C \\C\\{z_0, \\ldots, z_n\\} \\big/ \\left( \\frac{\\partial f}{\\partial z_0}, \\ldots, \\frac{\\partial f}{\\partial z_n} \\right).$$\nThen each fibre of $\\phi$ has the homotopy type of a [[wedge|5JSWM1HBX2]] of $\\mu$ spheres of dimension $n$.\n\n## Remarks\n\nThe fibration is the Milnor fibration, its fibre is the Milnor fibre, and $K$ is the link of the singularity.\nIn the isolated case $K$ is a smooth manifold of real dimension $2n - 1$, and Milnor showed that it is $(n-2)$-connected.\nExample: for $f = z_0^2 + z_1^3$ with $n = 1$, the link $K$ in $S^3$ is the trefoil knot, and $\\mu = 1 \\cdot 2 = 2$.\nThe Milnor fibre is a punctured torus, which has the homotopy type of a wedge of two circles.\nFor $f = z_0^2 + z_1^2$ the link is the Hopf link and the fibre is an annulus, homotopic to one circle, in agreement with $\\mu = 1$.\nThe theorem holds equally for germs of holomorphic functions at $0$.\nMilnor proved it in his book *Singular points of complex hypersurfaces*; the proof uses a curve selection lemma from real algebraic geometry.\n\n## Explanations\n\n### L1 human\n\nTake a complex polynomial with a singular point at the origin and look at it on a tiny sphere around that point.\nThe zero set cuts the sphere in a knot or link, such as the trefoil for $z_0^2 + z_1^3$.\nThe rest of the sphere is swept out by a circle's worth of pieces, one for each argument of $f$.\nAll of them have the same shape, and all are bounded by the link.\nThe theorem says that these pieces are as simple as possible up to homotopy: each is a bouquet of spheres, as many as the Milnor number.\n\n### L2 human\n\nThe Milnor fibration is the basic topological structure attached to an isolated hypersurface singularity.\nThe monodromy of the fibration acts on the middle homology of the fibre, a free abelian group of rank $\\mu$.\nIts eigenvalues and the Seifert form carry much of the topology of the singularity.\nFor $n = 1$ the link is an algebraic knot or link in $S^3$, and the fibre is a Seifert surface, so plane curve singularities give fibred links.\nThe Milnor number appearing in the theorem is also the dimension of the local algebra used in [[finite determinacy|EGPP69XA4F]] and the classification of [[simple critical points|M5437FP73G]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "0YTCMFB4HC"
    ],
    "kind": "statement",
    "msc": [
      "32S55",
      "58K10",
      "14B05"
    ],
    "name": "Milnor fibration theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Milnor_map"
      }
    ],
    "tag": "3CGMRT9W7J",
    "uses": [
      "CX4M8K61AB",
      "0YTCMFB4HC",
      "5JSWM1HBX2",
      "P9CM4GQ2T0"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/3CGMRT9W7J/",
  "links": [
    {
      "field": "about",
      "target": "0YTCMFB4HC"
    },
    {
      "field": "uses",
      "target": "CX4M8K61AB"
    },
    {
      "field": "uses",
      "target": "0YTCMFB4HC"
    },
    {
      "field": "uses",
      "target": "5JSWM1HBX2"
    },
    {
      "field": "uses",
      "target": "P9CM4GQ2T0"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/3CGMRT9W7J.md",
  "msc": [
    "32S55",
    "58K10",
    "14B05"
  ],
  "name": "Milnor fibration theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3CG/3CGMRT9W7J.md",
  "status": null,
  "tag": "3CGMRT9W7J",
  "uri": "m2://3CGMRT9W7J",
  "url": "https://tamnd.github.io/math2/wiki/milnor-fibration-theorem/",
  "writing": {
    "tag": "3CGMRT9W7J",
    "kind": "statement",
    "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
      }
    ]
  }
}