nLab
cohomology operation

Idea

A cohomology operation is a family of morphism between cohomology groups, which is natural with respect to the base space.

Equivalently, if the cohomology theory has a classifying space (as it does for all usual notions of cohomology, in particular for all generalized (Eilenberg-Steenrod) cohomology theories) then, by the Yoneda lemma, cohomology operations are in natural bijection with homotopy-classes of morphisms between classifying spaces.

(This statement is made fully explicit for instance below def. 12.3.22 in (Aguilar-Gitler-Prieto).

$$\{H^k(-,E)\to H^l(-,F)\}\simeq \mathrm{Ho}(E_k,F_l).$$\{ H^k(-, E) \to H^l(-, F) \} \simeq Ho(E_k, F_l) \,.

Examples

• Bockstein homomorphism

• Steenrod squares

• cup product

• Massey product

• Every universal characteristic class is a cohomology operation.

References

Steenrod’s original colloquium lectures were published as:

• Norman Steenrod, Cohomology operations, and obstructions to extending continuous functions Advances in Math. 8, 371–416. (1972). (scanned pdf)

Textbook accounts include the following.

• Robert E. Mosher and Martin C. Tangora, Cohomology Operations and Applications in Homotopy Theory, Harper and Row (1968)

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer (2002)