{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "AX4ZM9BV3Z"
    },
    {
      "field": "about",
      "target": "NWXJACYJC5"
    },
    {
      "field": "about",
      "target": "SGHMSBHHA0"
    },
    {
      "field": "uses",
      "target": "AX4ZM9BV3Z"
    },
    {
      "field": "uses",
      "target": "FFH90QRFFJ"
    },
    {
      "field": "uses",
      "target": "JGBW4KVMN8"
    },
    {
      "field": "uses",
      "target": "K99YMX66XD"
    },
    {
      "field": "uses",
      "target": "MKBSS15J3Z"
    },
    {
      "field": "uses",
      "target": "N685RXHDZX"
    },
    {
      "field": "uses",
      "target": "NWXJACYJC5"
    },
    {
      "field": "uses",
      "target": "SA493EH7DT"
    },
    {
      "field": "uses",
      "target": "SGHMSBHHA0"
    },
    {
      "field": "defines",
      "target": "DAXBFJ8G6T"
    }
  ],
  "body": "\n## Summary\n\nRelative homology assigns to a space $X$ and a subspace $A$ the groups $H_n(X, A)$.\nThey measure the homology of $X$ while ignoring everything that happens inside $A$.\nIt is built from the singular chains of $X$ modulo the chains of $A$.\nRelative groups fit into the long exact sequence of the pair and are the setting of the excision theorem.\nThis makes them the main tool for computing [[singular homology|M0Q5CFJ7ZH]].\n\n## Explanations\n\n### L1 human\n\nThink of $A$ as a region that is painted over and treated as if it were a single point.\nA relative cycle is a chain whose boundary lies in $A$, so it may have loose ends as long as they end inside $A$.\nFor the interval $X = [0, 1]$ with $A = \\set{0, 1}$, the path from $0$ to $1$ is a relative cycle, because its boundary consists of the two end points.\nIt represents a generator of $H_1([0,1], \\set{0,1}) \\cong \\Z$, just as if the two end points were glued into a circle.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "55N10"
    ],
    "name": "Relative homology",
    "same_as": {
      "wikidata": "Q1959911",
      "wikipedia": "Relative homology"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Relative_homology"
      }
    ],
    "tag": "4BVWH0BN28"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/4BVWH0BN28/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/4BVWH0BN28.md",
  "msc": [
    "55N10"
  ],
  "name": "Relative homology",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4BV/4BVWH0BN28.md",
  "status": null,
  "tag": "4BVWH0BN28",
  "uri": "m2://4BVWH0BN28",
  "url": "https://math2.tamnd.com/wiki/relative-homology/",
  "writing": {
    "tag": "4BVWH0BN28",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}