{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "2YPJ39STQM"
    },
    {
      "field": "about",
      "target": "7KEDZ9TFEE"
    },
    {
      "field": "about",
      "target": "GS8SCGYPX7"
    },
    {
      "field": "uses",
      "target": "2YPJ39STQM"
    },
    {
      "field": "uses",
      "target": "7KEDZ9TFEE"
    },
    {
      "field": "uses",
      "target": "GS8SCGYPX7"
    },
    {
      "field": "uses",
      "target": "HEV7170SEK"
    },
    {
      "field": "defines",
      "target": "H0T4RFG2AD"
    }
  ],
  "body": "\n## Summary\n\nThe Steenrod squares $\\operatorname{Sq}^i$, $i \\ge 0$, are [[cohomology operations|K8T15QHN3W]] on mod $2$ cohomology that raise degree by $i$.\nThey are additive, stable under suspension, and extend the cup square: $\\operatorname{Sq}^n x = x \\smile x$ when $x$ has degree $n$.\nThey generate the [[Steenrod algebra|JWTTXSGD0E]] and give the mod $2$ cohomology of a space the structure of a module over it.\nThey detect maps that cup products miss, such as the suspensions of the Hopf maps, and they compute [[Stiefel–Whitney classes|JQCK7A210W]] through [[Wu's formula|7KEDZ9TFEE]].\n\n## Explanations\n\n### L1 human\n\nIn mod $2$ cohomology, squaring a class with the cup product doubles its degree.\nSteenrod squares fill in the steps between: $\\operatorname{Sq}^i$ adds $i$ to the degree.\nThey start with the identity at $i = 0$ and end with the cup square when $i$ equals the degree.\nFor the complex projective plane, $\\operatorname{Sq}^2$ of the degree $2$ generator is the nonzero top class, and this shows that the Hopf map $S^3 \\to S^2$ is not null-homotopic.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "55S10",
      "55S05"
    ],
    "name": "Steenrod square",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Steenrod_algebra"
      }
    ],
    "specializes": [
      "K8T15QHN3W"
    ],
    "tag": "VR7RHP41M0"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/VR7RHP41M0/",
  "links": [
    {
      "field": "specializes",
      "target": "K8T15QHN3W"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/VR7RHP41M0.md",
  "msc": [
    "55S10",
    "55S05"
  ],
  "name": "Steenrod square",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VR7/VR7RHP41M0.md",
  "status": null,
  "tag": "VR7RHP41M0",
  "uri": "m2://VR7RHP41M0",
  "url": "https://math2.tamnd.com/wiki/steenrod-square/",
  "writing": {
    "tag": "VR7RHP41M0",
    "kind": "concept",
    "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
      }
    ]
  }
}