Steenrod squares from cup-i

Let $n\ge 0$, let $x\in H^n(X;{\mathbb{F}}_2)$, and choose a cocycle $a\in C^n(X;{\mathbb{F}}_2)$ representing $x$. For $0\le k\le n$, define

$$S{q}^k(x):=[a{⌣}_{n-k}a]\in H^{n+k}(X;{\mathbb{F}}_2),$$

using Higher cup-i products. Define $S{q}^k(x)=0$ for $k<0$ or $k>n$. The identical formula defines relative squares on $H^n(X,A;{\mathbb{F}}_2)$.

The displayed cochain is a cocycle. Indeed, the identity from Cup-i coboundary identity and $\delta a=0$ give

$$\delta (a{⌣}_{n-k}a)=a{⌣}_{n-k-1}a+a{⌣}_{n-k-1}a=0$$

in characteristic two, including $k=n$ because ${⌣}_{-1}=0$. Thus the formula at least determines a cohomology class. Independence of the cocycle and of the fixed higher-diagonal system is proved next. The definition is choice-free once those data are fixed; on empty or one-point reduced cohomology, and on the zero class, it gives zero.