nLab
cohomology operation

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Homotopy theory

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

Examples

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)

Revised on September 14, 2014 07:33:38 by Tim Porter (2.26.35.200)