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A cohomology operation is stable when it commutes with the suspension isomorphism on reduced cohomology:Products of positive-degree reduced classes on a suspension vanish, but stable operations can still connect its nonzero cohomology classes. The Steenrod squares and, at odd primes, Steenrod reduced powers are examples.

Let $x\in H^2({\mathbb{CP}}^2;{\mathbb{F}}_2)$ be the hyperplane class modulo two. It satisfies ${\mathrm{Sq}}^{2}x=x^2\ne 0$. Stability gives ${\mathrm{Sq}}^2(\sigma x)=\sigma (x^2)\ne 0$ on $\Sigma {\mathbb{CP}}^2$. The suspension of $S^2\vee S^4$ has the same additive groups and vanishing positive-degree products, but its corresponding degree-three class has zero ${\mathrm{Sq}}^2$, by pullback from $S^3$. Thus these suspensions have different cohomology operations even though their additive groups and cup products agree.^[ref]

The Steenrod algebra organizes stable mod-$p$ cohomology operations. At $p=2$ it is generated by the Steenrod squares; at an odd prime it is generated by the Steenrod reduced powers and Bockstein homomorphism. Cohomology is a module over this algebra, and every continuous map induces a homomorphism preserving its action.

The Steenrod squares are natural mod-two cohomology operationsThey satisfy ${\mathrm{Sq}}^0=1$, ${\mathrm{Sq}}^{i}a=0$ for $i>n$, and the top-square identity ${\mathrm{Sq}}^{n}a=a⌣a$. The Cartan formula computes squares of products, while the Adem relations constrain compositions. They are stable cohomology operations, so they retain information after suspension.

For the degree-one generator $x$ of the mod-two cohomology ring of real projective space, the Cartan formula giveswith coefficients modulo two and powers above the space's dimension set to zero. For example, ${\mathrm{Sq}}^1(x^3)=x^4$ and ${\mathrm{Sq}}^2(x^3)=x^5$ when those degrees exist. The operation ${\mathrm{Sq}}^1$ is the mod-two Bockstein homomorphism. The higher squares therefore extend familiar cup-square and coefficient-sequence information into a natural family of finer information preserved by homotopy equivalences.^[ref]