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  "body": "\n## Definition\n\nLet $p$ be an odd prime, and write $H^n(X)$ for the [[singular cohomology|6N7BC70CX0]] group $H^n(X; \\F_p)$ of a space $X$.\nThe *Steenrod reduced powers* are the unique family of homomorphisms\n$$P^i \\colon H^n(X) \\to H^{n + 2i(p-1)}(X), \\qquad i \\ge 0,\\ n \\ge 0,$$\ndefined for all spaces $X$, with the following properties.\n\n1. Each $P^i$ is natural: $f^* \\circ P^i = P^i \\circ f^*$ for every continuous map $f \\colon Y \\to X$.\n2. Each $P^i$ commutes with the suspension isomorphism $H^n(X) \\cong H^{n+1}(SX)$ in reduced cohomology.\n3. $P^0$ is the identity.\n4. If $x \\in H^{2i}(X)$, then $P^i x = x^p$, the $p$-th [[cup product|JX47MS1R1H]] power of $x$.\n5. If $x \\in H^n(X)$ and $2i \u003e n$, then $P^i x = 0$.\n6. (Cartan formula) $P^k(x \\smile y) = \\sum_{i + j = k} P^i x \\smile P^j y$ for all $x, y \\in H^*(X)$.\n\n## Remarks\n\nExistence of such a family is a theorem of Steenrod; uniqueness follows from the axioms by evaluating on products of Eilenberg–MacLane spaces.\nThe operations also exist on cohomology of pairs and commute with the coboundary maps of pairs.\n\nExample: on $X = \\C P^\\infty$, with $H^*(X) = \\F_p[x]$ and $\\deg x = 2$, clauses 4 and 6 give $P^i(x^m) = \\binom{m}{i} x^{m + i(p-1)}$.\nFor instance $P^1 x = x^p$, which is nonzero.\n\nNon-example: let $Q^0$ be the identity and $Q^i = 0$ for $i \u003e 0$.\nThis family satisfies clauses 1, 2, 3, 5 and 6.\nIt fails clause 4: on $\\C P^\\infty$ one has $Q^1 x = 0$, while $x^p \\neq 0$.\n\nThe operation $P^i$ is a [[cohomology operation|K8T15QHN3W]] of type $(\\F_p, n; \\F_p, n + 2i(p-1))$.\nFor $p = 2$ the analogous operations are the Steenrod squares $Sq^i$, with $Sq^{2i}$ playing the role of $P^i$.\n",
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