{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "WWX8HCN260"
    }
  ],
  "body": "\n## Statement\n\nLet $\\pi \\colon E \\to B$ be a [[Serre fibration|PB9JXQ2KC3]] with $B$ path-connected, let $b_0 \\in B$, and let $F = \\pi^{-1}(b_0)$ be the fibre.\nLet $R$ be a commutative ring.\nThen there is a first-quadrant cohomological spectral sequence $(E_r^{p,q}, d_r)_{r \\ge 2}$, with $d_r \\colon E_r^{p,q} \\to E_r^{p+r, q-r+1}$, such that:\n\n1. $E_2^{p,q} = H^p(B; \\mathcal{H}^q(F; R))$, the [[singular cohomology|6N7BC70CX0]] of $B$ with [[local coefficients|M58WRZ7PFM]] in the system $b \\mapsto H^q(\\pi^{-1}(b); R)$;\n2. the spectral sequence converges to $H^{p+q}(E; R)$: there is a filtration of $H^n(E; R)$ whose successive quotients are the groups $E_\\infty^{p, n-p}$;\n3. it is multiplicative: each $E_r$ is a bigraded ring and $d_r$ is a derivation.\nOn $E_2^{p,q} \\otimes E_2^{s,t}$ the product is the cup product times $(-1)^{qs}$.\n\nIf the [[fundamental group|S1XK3HXHFX]] $\\pi_1(B, b_0)$ acts as the identity on $H^*(F; R)$, for instance if $B$ is simply connected, then $E_2^{p,q} = H^p(B; H^q(F; R))$ with constant coefficients.\n\n## Remarks\n\nOne proof filters the singular cochains of $E$ by the preimages of the skeleta of a CW approximation of $B$.\nAnother proof filters the singular simplices of $E$ by how they project to $B$.\nThere is a homological version with $E^2_{p,q} = H_p(B; \\mathcal{H}_q(F; R))$ converging to $H_{p+q}(E; R)$.\n\nExample: for $n \\ge 2$, the [[path space fibration|D3BN7MBKTR]] $\\Omega S^n \\to P S^n \\to S^n$ has contractible total space.\nSo all classes except $E_2^{0,0}$ must die, which forces $d_n \\colon E_n^{0, k(n-1)} \\to E_n^{n, (k-1)(n-1)}$ to be isomorphisms.\nIt follows that $H^i(\\Omega S^n; \\Z) \\cong \\Z$ when $n - 1$ divides $i$ and $0$ otherwise.\n\n## Explanations\n\n### L1 human\n\nA fibration is a space $E$ that looks locally like a product of a base $B$ and a fibre $F$.\nThe Serre spectral sequence starts from the cohomology of $B$ with coefficients in the cohomology of $F$, as if $E$ were a product.\nThen it corrects this guess step by step, and the end result is the cohomology of $E$.\n\n### L2 human\n\nThe Serre spectral sequence is the main tool for computing cohomology of fibre bundles, loop spaces, homogeneous spaces and Eilenberg–MacLane spaces.\nIt gives the Gysin and Wang sequences as special cases when the fibre or the base is a sphere.\nSerre used it to prove that the homotopy groups of spheres are finitely generated, and finite except in a few degrees.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "PB9JXQ2KC3"
    ],
    "kind": "statement",
    "msc": [
      "55T10",
      "55R20"
    ],
    "name": "Serre spectral sequence",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7455410",
      "wikipedia": "Serre spectral sequence"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Serre_spectral_sequence"
      }
    ],
    "tag": "AHW4B5480K",
    "uses": [
      "PB9JXQ2KC3",
      "M58WRZ7PFM",
      "6N7BC70CX0",
      "S1XK3HXHFX",
      "D3BN7MBKTR"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/AHW4B5480K/",
  "links": [
    {
      "field": "about",
      "target": "PB9JXQ2KC3"
    },
    {
      "field": "uses",
      "target": "PB9JXQ2KC3"
    },
    {
      "field": "uses",
      "target": "M58WRZ7PFM"
    },
    {
      "field": "uses",
      "target": "6N7BC70CX0"
    },
    {
      "field": "uses",
      "target": "S1XK3HXHFX"
    },
    {
      "field": "uses",
      "target": "D3BN7MBKTR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/AHW4B5480K.md",
  "msc": [
    "55T10",
    "55R20"
  ],
  "name": "Serre spectral sequence",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AHW/AHW4B5480K.md",
  "status": null,
  "tag": "AHW4B5480K",
  "uri": "m2://AHW4B5480K",
  "url": "https://math2.tamnd.com/wiki/serre-spectral-sequence/",
  "writing": {
    "tag": "AHW4B5480K",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}