Contents

cohomology

# Contents

## 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).

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

## 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.

Discussion of refinement of cohomology operations to differential cohomology is in

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