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Norman Steenrod developed operations on cohomology that reveal structure beyond the cohomology groups alone. His original work constructs the mod-two Steenrod squares, organized under composition in the Steenrod algebra.^[ref]

An operation $S{q}^i$ sends a class in degree $n$ to one in degree $n+i$ and behaves naturally under continuous maps. The top square satisfies $S{q}^n(a)=a⌣a$, connecting the operations with the cup product. Such compatibility makes them useful for distinguishing spaces and detecting restrictions on maps, even when simpler algebraic invariants agree.