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:

Review in:

• Peter May, chapter 22, section 5 of: A concise course in algebraic topology, University of Chicago Press 1999 (ISBN:978-0226511832, pdf)

Textbook accounts include the following.