{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $n \\ge 1$ and let $a_0, \\dots, a_n \\ge 2$ be integers.\nThe *Brieskorn manifold* is\n$$\\Sigma(a_0, \\dots, a_n) = \\set{z \\in \\C^{n+1} : z_0^{a_0} + z_1^{a_1} + \\cdots + z_n^{a_n} = 0} \\cap S^{2n+1},$$\nwhere $S^{2n+1}$ is the unit sphere in $\\C^{n+1}$.\nIt is the [[link|C831AN3RFM]] at $0$ of the hypersurface $f^{-1}(0)$, where $f = z_0^{a_0} + \\cdots + z_n^{a_n}$.\n\nCheck: the partial derivatives of $f$ are $a_j z_j^{a_j - 1}$, which vanish together only at $z = 0$.\nSo $f$ has an [[isolated singularity|VTGYEES3VS]] at $0$ and $f^{-1}(0)$ is smooth away from $0$.\nSince $f$ is weighted homogeneous, the sphere of radius $1$ gives the same link as any small sphere.\nSo $\\Sigma(a_0, \\dots, a_n)$ is a smooth closed manifold of dimension $2n - 1$.\nThe [[Milnor number|1ZE51P7GRQ]] of $f$ is $(a_0 - 1)(a_1 - 1) \\cdots (a_n - 1)$, because the monomials $z_0^{b_0} \\cdots z_n^{b_n}$ with $0 \\le b_j \\le a_j - 2$ form a basis of the local algebra.\n\n## Remarks\n\n$\\Sigma(2, 3, 5)$ is the [[Poincaré homology sphere|NHERCK7C43]], a $3$-manifold with the homology of $S^3$ but a nontrivial fundamental group of order $120$.\nFor $n = 1$, $\\Sigma(p, q)$ is the torus link of type $(p, q)$ in $S^3$, a knot when $p$ and $q$ are coprime.\nBrieskorn showed that the $7$-manifolds $\\Sigma(6k - 1, 3, 2, 2, 2)$ for $k = 1, \\dots, 28$ are homeomorphic to $S^7$ and realize the $28$ oriented diffeomorphism classes of [[exotic spheres|6WYEMXSHD9]] of dimension $7$.\n\n## Explanations\n\n### L1 human\n\nBrieskorn manifolds come from the simplest polynomials one can write: sums of powers of the variables.\nCut the zero set of such a polynomial with the unit sphere, and the result is a smooth manifold.\nChanging the exponents changes the manifold in a computable way.\nSome choices give spheres that are equal to the usual sphere as topological spaces but not as smooth manifolds.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "C831AN3RFM"
    ],
    "kind": "example",
    "msc": [
      "32S55",
      "57R60",
      "32S25"
    ],
    "name": "Brieskorn manifold",
    "same_as": {
      "wikidata": "Q4967112",
      "wikipedia": "Brieskorn manifold"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Brieskorn_manifold"
      }
    ],
    "tag": "5E9E5R9RYZ",
    "uses": [
      "VTGYEES3VS",
      "1ZE51P7GRQ",
      "6WYEMXSHD9",
      "NHERCK7C43"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/5E9E5R9RYZ/",
  "links": [
    {
      "field": "instance_of",
      "target": "C831AN3RFM"
    },
    {
      "field": "uses",
      "target": "VTGYEES3VS"
    },
    {
      "field": "uses",
      "target": "1ZE51P7GRQ"
    },
    {
      "field": "uses",
      "target": "6WYEMXSHD9"
    },
    {
      "field": "uses",
      "target": "NHERCK7C43"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5E9E5R9RYZ.md",
  "msc": [
    "32S55",
    "57R60",
    "32S25"
  ],
  "name": "Brieskorn manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5E9/5E9E5R9RYZ.md",
  "status": null,
  "tag": "5E9E5R9RYZ",
  "uri": "m2://5E9E5R9RYZ",
  "url": "https://tamnd.github.io/math2/wiki/brieskorn-manifold/",
  "writing": {
    "tag": "5E9E5R9RYZ",
    "kind": "example",
    "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
      }
    ]
  }
}