 Steenrod square

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]