Abstract

Cohomology operations are absolutely essential in making cohomology an effective tool for studying spaces. In particular, the mod-p cohomology groups of a space X are enhanced with a binary cup product, a Bockstein derivation, and Steenrod’s reduced power operations; these satisfy relations such as graded-commutativity, the Cartan formula, the Adem relations, and the instability relations [90]. The combined structure of these cohomology operations is very effective in homotopy theory because of three critical properties.

These operations are natural. We can exclude the possibility of certain maps between spaces because they would not respect these operations.

These operations are constrained. We can exclude the existence of certain spaces because the cup product and power operations would be incompatible with the relations that must hold.

These operations are complete. Because cohomology is representable, we can determine all possible natural operations which take an n-tuple of cohomology elements and produce a new one. All operations are built, via composition, from these basic operations. All relations between these operations are similarly built from these basic relations.