{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer.\nA *homology $n$-sphere* is a closed (compact, without boundary) [[topological $n$-manifold|QQQ9B1DK4P]] $X$ whose integral [[singular homology|M0Q5CFJ7ZH]] is that of the [[$n$-sphere|T62369RFH6]]:\n$$H_i(X; \\Z) \\cong \\begin{cases} \\Z \u0026 \\text{if } i = 0 \\text{ or } i = n, \\\\ 0 \u0026 \\text{otherwise.} \\end{cases}$$\n\n## Remarks\n\nThe condition $H_0(X; \\Z) \\cong \\Z$ says that $X$ is connected, and $H_n(X; \\Z) \\cong \\Z$ says that $X$ is orientable.\n\nExample: the Poincaré homology sphere $P = S^3 / I^\\ast$, where $I^\\ast \\subset S^3$ is the binary icosahedral group of order $120$ acting by left multiplication.\nIts fundamental group $I^\\ast$ is perfect, so $H_1(P; \\Z)$, the abelianisation of $I^\\ast$, is $0$; Poincaré duality and the universal coefficient theorem then give $H_2(P; \\Z) = 0$ and $H_3(P; \\Z) \\cong \\Z$.\n\nNon-example: the lens space $L(p, q)$ with $p \\ge 2$ is a closed orientable $3$-manifold, but $H_1(L(p, q); \\Z) \\cong \\Z/p \\neq 0$.\n\nA simply connected homology $n$-sphere is homotopy equivalent to $S^n$, by the Hurewicz and Whitehead theorems.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "HK8NS02NNE",
    "kind": "definition",
    "msc": [
      "57N65",
      "57K30"
    ],
    "name": "Homology sphere",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Homology_sphere"
      }
    ],
    "tag": "SJ9AMSQA7P",
    "uses": [
      "QQQ9B1DK4P",
      "M0Q5CFJ7ZH",
      "T62369RFH6"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/SJ9AMSQA7P/",
  "links": [
    {
      "field": "defines",
      "target": "HK8NS02NNE"
    },
    {
      "field": "uses",
      "target": "QQQ9B1DK4P"
    },
    {
      "field": "uses",
      "target": "M0Q5CFJ7ZH"
    },
    {
      "field": "uses",
      "target": "T62369RFH6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/SJ9AMSQA7P.md",
  "msc": [
    "57N65",
    "57K30"
  ],
  "name": "Homology sphere",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SJ9/SJ9AMSQA7P.md",
  "status": null,
  "tag": "SJ9AMSQA7P",
  "uri": "m2://SJ9AMSQA7P",
  "url": "https://math2.tamnd.com/wiki/homology-sphere-definition/",
  "writing": {
    "tag": "SJ9AMSQA7P",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}