Algebraic topology Cohomology operations and obstruction theory Theorem

Adem relations

Builds on Steenrod square Steenrod reduced power Bockstein homomorphism Binomial coefficient

Statement

Let $p$ be a prime and consider operations on $H^*(X; \F_p)$ for spaces $X$ . All binomial coefficients below are read modulo $p$ , and $\binom{m}{k} = 0$ when $k < 0$ or $k > m$ .

For $p = 2$ , the Steenrod squares satisfy, for all integers $a, b > 0$ with $a < 2b$ , $Sq^a Sq^b = \sum_{j=0}^{\floor{a/2}} \binom{b - 1 - j}{a - 2j} Sq^{a + b - j} Sq^j.$
For $p$ odd, the Steenrod reduced powers satisfy, for all integers $a, b > 0$ with $a < pb$ , $P^a P^b = \sum_{j=0}^{\floor{a/p}} (-1)^{a + j} \binom{(p-1)(b - j) - 1}{a - pj} P^{a + b - j} P^j.$
For $p$ odd, with $\beta$ the Bockstein homomorphism of $0 \to \Z/p \to \Z/p^2 \to \Z/p \to 0$ , for all integers $a, b > 0$ with $a \le pb$ , $P^a \beta P^b = \sum_{j=0}^{\floor{a/p}} (-1)^{a + j} \binom{(p-1)(b - j)}{a - pj} \beta P^{a + b - j} P^j + \sum_{j=0}^{\floor{(a-1)/p}} (-1)^{a + j + 1} \binom{(p-1)(b - j) - 1}{a - pj - 1} P^{a + b - j} \beta P^j.$

Remarks

One proof computes the operations on the cohomology of a product of Eilenberg–MacLane spaces, or of symmetric products, where all compositions are visible. Another proof packages the operations into a generating function and uses the Cartan formula.

Example for $p = 2$ : taking $a = b = 1$ gives $Sq^1 Sq^1 = \binom{0}{1} Sq^2 = 0$ . Taking $a = 1, b = 2$ gives $Sq^1 Sq^2 = Sq^3$ . Taking $a = b = 2$ gives $Sq^2 Sq^2 = Sq^3 Sq^1$ .

The condition $a < 2b$ is needed: when $a \ge 2b$ the composite $Sq^a Sq^b$ is already admissible and is not rewritten. The relations, together with $Sq^0 = 1$ (and $P^0 = 1$ , $\beta^2 = 0$ for odd $p$ ), give all relations among composites of these operations. So the mod $p$ Steenrod algebra is the free associative algebra on the operations modulo these relations.

Going deeper

For a student

The Adem relations make the Steenrod algebra computable: the admissible monomials form a basis, which is the Cartan–Serre basis. They show that $Sq^n$ is decomposable unless $n$ is a power of $2$ , so the algebra is generated by the $Sq^{2^k}$ . This fact is the starting point of the Hopf invariant one problem and of the Adams spectral sequence.

Reading path

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Tag	`GS8SCGYPX7`
URI	`m2://GS8SCGYPX7`
Kind	statement, theorem
Writing	W0 stub
MSC	55S10
Status	This copy of the site has no build data. Run `m2 status GS8SCGYPX7`.
About	Steenrod square, Steenrod reduced power
Uses	Steenrod square, Steenrod reduced power, Bockstein homomorphism, Binomial coefficient
Changed	2 November 2026

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