 Stable cohomology operation

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]