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.

• 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)

• Peter May, chapter 22, section 5 of A concise course in algebraic topology (pdf)

Revised on May 6, 2015 10:35:19 by Urs Schreiber (195.113.30.252)