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  "body": "\n## Summary\n\nFor an odd prime $p$, the Steenrod reduced powers $P^i$ are [[cohomology operations|K8T15QHN3W]] on mod $p$ [[singular cohomology|6N7BC70CX0]] that raise degree by $2i(p-1)$.\nThey play for odd primes the role that the Steenrod squares play for the prime $2$.\nTogether with the Bockstein homomorphism they generate all stable mod $p$ cohomology operations.\nThey detect maps that ordinary cohomology groups and [[cup products|JX47MS1R1H]] cannot see.\n\n## Explanations\n\n### L1 human\n\nCohomology with coefficients in $\\F_p$ already has a product, so a class $x$ has a $p$-th power $x^p$.\nThe reduced power $P^i$ is a natural way to produce new classes from $x$ that interpolate between $x$ itself and $x^p$.\nIf $x$ has degree $2i$, then $P^i x = x^p$, and if the degree of $x$ is less than $2i$, then $P^i x = 0$.\nFor example, on $\\C P^\\infty$ with generator $x$ of degree $2$, one has $P^1 x = x^p$.\nBecause these operations commute with every map of spaces, a space where they act differently from another space cannot be homotopy equivalent to it.\n",
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