Steenrod square

Builds on Cohomology operation Singular cohomology Cup product

Definition

Write $H^*(X, A)$ for singular cohomology of a pair of spaces with coefficients in $\Z/2\Z$ . The Steenrod squares are the cohomology operations $\operatorname{Sq}^i \colon H^n(X, A) \to H^{n+i}(X, A), \quad i \ge 0, \ n \ge 0,$ defined for all pairs $(X, A)$ , that satisfy:

Naturality: $f^* \operatorname{Sq}^i = \operatorname{Sq}^i f^*$ for every map of pairs $f$ .
Additivity: $\operatorname{Sq}^i(x + y) = \operatorname{Sq}^i x + \operatorname{Sq}^i y$ .
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.
Stability: $\operatorname{Sq}^i$ commutes with the connecting homomorphism $\delta \colon H^n(A) \to H^{n+1}(X, A)$ .
$\operatorname{Sq}^0$ is the identity.
If $x$ has degree $n$ , then $\operatorname{Sq}^n x = x \smile x$ , and $\operatorname{Sq}^i x = 0$ for $i > n$ .

There is exactly one family of operations with these properties.

Remarks

The total square is $\operatorname{Sq} = \operatorname{Sq}^0 + \operatorname{Sq}^1 + \operatorname{Sq}^2 + \cdots$ ; on each class only finitely many terms are nonzero. The square $\operatorname{Sq}^1$ is the Bockstein homomorphism of the sequence $0 \to \Z/2\Z \to \Z/4\Z \to \Z/2\Z \to 0$ .

Example: 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. By the Cartan formula, $\operatorname{Sq}(a^n) = (a + a^2)^n$ , so $\operatorname{Sq}^i(a^n) = \binom{n}{i} a^{n+i}$ . For instance $\operatorname{Sq}^1(a^2) = 2a^3 = 0$ and $\operatorname{Sq}^1(a^3) = a^4$ .

Non-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 $H^2(\R P^\infty)$ .

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Tag	`H0T4RFG2AD`
URI	`m2://H0T4RFG2AD`
Kind	definition
Writing	W0 stub
MSC	55S10, 55S05
Defines	Steenrod square
Uses	Cohomology operation, Singular cohomology, Cup product
Changed	2 November 2026

Steenrod square

Definition

Remarks

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