{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a topological space and let\n$$0 \\to A \\xrightarrow{i} B \\xrightarrow{j} C \\to 0$$\nbe a short [[exact sequence|CBDM280TDF]] of [[abelian groups|071DDFVG4J]].\nLet $C^*(X; G) = \\Hom(C_*(X), G)$ be the singular cochain complex with coefficients in $G$.\nSince each group $C_n(X)$ of singular chains is free abelian, the induced sequence of cochain complexes\n$$0 \\to C^*(X; A) \\to C^*(X; B) \\to C^*(X; C) \\to 0$$\nis exact.\nThe *Bockstein homomorphism* is its connecting homomorphism\n$$\\beta \\colon H^n(X; C) \\to H^{n+1}(X; A)$$\nin [[singular cohomology|6N7BC70CX0]].\nExplicitly, let $c$ be a cocycle representing a class in $H^n(X; C)$, and choose a cochain $b$ with values in $B$ with $j \\circ b = c$.\nThen $\\delta b = i \\circ a$ for a unique cocycle $a$, and $\\beta[c] = [a]$.\n\n## Remarks\n\nThe same construction on chains gives a homology Bockstein $H_n(X; C) \\to H_{n-1}(X; A)$ in [[homology with coefficients|XT7VWNTKGB]].\nThe most used case is $0 \\to \\Z/p \\to \\Z/p^2 \\to \\Z/p \\to 0$, which gives a natural map $\\beta \\colon H^n(X; \\F_p) \\to H^{n+1}(X; \\F_p)$ with $\\beta \\circ \\beta = 0$.\n\nExample: let $X = \\R P^2$ and $p = 2$, so $H^1(X; \\Z/2) \\cong H^2(X; \\Z/2) \\cong \\Z/2$, while $H^1(X; \\Z/4) \\cong \\Z/2$ and $H^2(X; \\Z/4) \\cong \\Z/2$.\nThe map $H^1(X; \\Z/4) \\to H^1(X; \\Z/2)$ is zero, since the class of order two in $H^1(X; \\Z/4)$ reduces to zero mod $2$.\nSo by exactness the generator of $H^1(X; \\Z/2)$ has nonzero Bockstein, which is the generator of $H^2(X; \\Z/2)$.\n\nNon-example of a nonzero Bockstein: for the split sequence $0 \\to \\Z/2 \\to \\Z/2 \\oplus \\Z/2 \\to \\Z/2 \\to 0$, every class lifts, and the Bockstein is the zero map.\nIn general $\\beta = 0$ whenever the coefficient sequence splits.\n\nFor $p = 2$ the Bockstein on mod $2$ cohomology equals the Steenrod square $Sq^1$.\nFor the sequence $0 \\to \\Z \\xrightarrow{p} \\Z \\to \\Z/p \\to 0$ one gets the integral Bockstein $H^n(X; \\Z/p) \\to H^{n+1}(X; \\Z)$, whose image consists of $p$-torsion classes.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "H7PCM776GR",
    "kind": "definition",
    "msc": [
      "55S05",
      "55N10"
    ],
    "name": "Bockstein homomorphism",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bockstein_homomorphism"
      }
    ],
    "tag": "VFPP3RZGK3",
    "uses": [
      "CBDM280TDF",
      "071DDFVG4J",
      "6N7BC70CX0",
      "XT7VWNTKGB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/VFPP3RZGK3/",
  "links": [
    {
      "field": "defines",
      "target": "H7PCM776GR"
    },
    {
      "field": "uses",
      "target": "CBDM280TDF"
    },
    {
      "field": "uses",
      "target": "071DDFVG4J"
    },
    {
      "field": "uses",
      "target": "6N7BC70CX0"
    },
    {
      "field": "uses",
      "target": "XT7VWNTKGB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/VFPP3RZGK3.md",
  "msc": [
    "55S05",
    "55N10"
  ],
  "name": "Bockstein homomorphism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VFP/VFPP3RZGK3.md",
  "status": null,
  "tag": "VFPP3RZGK3",
  "uri": "m2://VFPP3RZGK3",
  "url": "https://math2.tamnd.com/wiki/bockstein-homomorphism-definition/",
  "writing": {
    "tag": "VFPP3RZGK3",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}