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  "body": "\n## Definition\n\nWrite $H^*(X, A)$ for [[singular cohomology|6N7BC70CX0]] of a pair of spaces with coefficients in $\\Z/2\\Z$.\nThe *Steenrod squares* are the [[cohomology operations|K8T15QHN3W]]\n$$\\operatorname{Sq}^i \\colon H^n(X, A) \\to H^{n+i}(X, A), \\quad i \\ge 0, \\ n \\ge 0,$$\ndefined for all pairs $(X, A)$, that satisfy:\n\n1. Naturality: $f^* \\operatorname{Sq}^i = \\operatorname{Sq}^i f^*$ for every map of pairs $f$.\n2. Additivity: $\\operatorname{Sq}^i(x + y) = \\operatorname{Sq}^i x + \\operatorname{Sq}^i y$.\n3. Cartan formula: $\\operatorname{Sq}^k(x \\smile y) = \\sum_{i + j = k} \\operatorname{Sq}^i x \\smile \\operatorname{Sq}^j y$, with $\\smile$ the [[cup product|JX47MS1R1H]].\n4. Stability: $\\operatorname{Sq}^i$ commutes with the connecting homomorphism $\\delta \\colon H^n(A) \\to H^{n+1}(X, A)$.\n5. $\\operatorname{Sq}^0$ is the identity.\n6. If $x$ has degree $n$, then $\\operatorname{Sq}^n x = x \\smile x$, and $\\operatorname{Sq}^i x = 0$ for $i \u003e n$.\n\nThere is exactly one family of operations with these properties.\n\n## Remarks\n\nThe total square is $\\operatorname{Sq} = \\operatorname{Sq}^0 + \\operatorname{Sq}^1 + \\operatorname{Sq}^2 + \\cdots$; on each class only finitely many terms are nonzero.\nThe square $\\operatorname{Sq}^1$ is the [[Bockstein homomorphism|H7PCM776GR]] of the sequence $0 \\to \\Z/2\\Z \\to \\Z/4\\Z \\to \\Z/2\\Z \\to 0$.\n\nExample: in $H^*(\\R P^\\infty) = (\\Z/2\\Z)[a]$ with $\\deg a = 1$, one has $\\operatorname{Sq} a = a + a^2$ by clauses 5 and 6.\nBy the Cartan formula, $\\operatorname{Sq}(a^n) = (a + a^2)^n$, so $\\operatorname{Sq}^i(a^n) = \\binom{n}{i} a^{n+i}$.\nFor instance $\\operatorname{Sq}^1(a^2) = 2a^3 = 0$ and $\\operatorname{Sq}^1(a^3) = a^4$.\n\nNon-example: the family with $\\operatorname{Sq}^0 = \\id$ and $\\operatorname{Sq}^i = 0$ for all $i \\ge 1$ satisfies clauses 1 to 5 but fails clause 6, since $\\operatorname{Sq}^1 a$ would be $0$ instead of $a^2$ in $H^2(\\R P^\\infty)$.\n",
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