Steenrod square

Summary

The Steenrod squares $\operatorname{Sq}^i$, $i \ge 0$, are cohomology operations on mod $2$ cohomology that raise degree by $i$. They are additive, stable under suspension, and extend the cup square: $\operatorname{Sq}^n x = x \smile x$ when $x$ has degree $n$. They generate the Steenrod algebra and give the mod $2$ cohomology of a space the structure of a module over it. They detect maps that cup products miss, such as the suspensions of the Hopf maps, and they compute Stiefel–Whitney classes through Wu's formula.