{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $K$ be a nonempty, [[compact|0TVJZYB6SH]], locally contractible proper subspace of the [[sphere|T62369RFH6]] $S^n$.\nThen for every integer $i$ there is an isomorphism\n$$\\tilde H_i(S^n \\setminus K; \\Z) \\cong \\tilde H^{\\,n-i-1}(K; \\Z),$$\nwhere $\\tilde H_i$ is [[reduced homology|NJQMWVK3QK]] and $\\tilde H^{\\,j}$ is reduced [[singular cohomology|6N7BC70CX0]].\n\n## Remarks\n\nThe proof uses a duality between the cohomology of a compact subset of a manifold and the homology of the manifold relative to its complement.\nThis duality is proved like [[Poincaré duality|9VY90HJVZX]].\nThe long exact sequence of the pair $(S^n, S^n \\setminus K)$ then gives the result.\nFor an arbitrary compact $K$, the statement still holds with Čech cohomology in place of singular cohomology.\nWith singular cohomology, local contractibility cannot be dropped: the Warsaw circle in $S^2$ has the singular cohomology of a point but separates $S^2$ into two components.\n\nExample: if $K \\subset S^n$ is homeomorphic to $S^{n-1}$, then $\\tilde H_0(S^n \\setminus K) \\cong \\tilde H^{\\,n-1}(S^{n-1}) \\cong \\Z$.\nSo the complement has exactly two path components, which is the Jordan–Brouwer separation theorem.\n\nExample: for a knot $K \\subset S^3$, that is, a subspace homeomorphic to $S^1$, $H_1(S^3 \\setminus K) \\cong \\tilde H^1(S^1) \\cong \\Z$, whatever the knot is.\n\n## Explanations\n\n### L1 human\n\nThe holes in the complement of a shape in a sphere are determined by the holes of the shape itself, with dimensions flipped.\nA circle drawn on the $2$-sphere cuts it into two pieces.\nA knotted circle in $3$-space always has a complement with exactly one independent loop around it, however complicated the knot.\n\n### L2 human\n\nAlexander duality computes the homology of complements of embedded subsets, such as knot complements and complements of embedded spheres.\nIt gives a quick proof of the Jordan–Brouwer separation theorem in all dimensions.\nIts proof is a direct application of the duality theory of manifolds, and its Čech version is a standard reason to introduce Čech cohomology.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "NJQMWVK3QK",
      "6N7BC70CX0"
    ],
    "kind": "statement",
    "msc": [
      "55M05"
    ],
    "name": "Alexander duality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q4720484",
      "wikipedia": "Alexander duality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Alexander_duality"
      }
    ],
    "tag": "KK05CSTM15",
    "uses": [
      "NJQMWVK3QK",
      "6N7BC70CX0",
      "T62369RFH6",
      "0TVJZYB6SH",
      "9VY90HJVZX"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/KK05CSTM15/",
  "links": [
    {
      "field": "about",
      "target": "NJQMWVK3QK"
    },
    {
      "field": "about",
      "target": "6N7BC70CX0"
    },
    {
      "field": "uses",
      "target": "NJQMWVK3QK"
    },
    {
      "field": "uses",
      "target": "6N7BC70CX0"
    },
    {
      "field": "uses",
      "target": "T62369RFH6"
    },
    {
      "field": "uses",
      "target": "0TVJZYB6SH"
    },
    {
      "field": "uses",
      "target": "9VY90HJVZX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/KK05CSTM15.md",
  "msc": [
    "55M05"
  ],
  "name": "Alexander duality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KK0/KK05CSTM15.md",
  "status": null,
  "tag": "KK05CSTM15",
  "uri": "m2://KK05CSTM15",
  "url": "https://math2.tamnd.com/wiki/alexander-duality/",
  "writing": {
    "tag": "KK05CSTM15",
    "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
      }
    ]
  }
}