Cohomology operation

Summary

A cohomology operation turns cohomology classes of one fixed degree into classes of another degree. It does so on every space at once, compatibly with all continuous maps. Examples are the cup square $x \mapsto x \smile x$, the Bockstein homomorphism and the Steenrod squares. Operations give extra structure on cohomology that can distinguish spaces and maps with the same cohomology groups or rings. By representability of cohomology, operations from degree $n$ with coefficients $G$ correspond to cohomology classes of the Eilenberg–MacLane space $K(G, n)$.