nLab
cohomology operation

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

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.

Discussion of refinement of cohomology operations to differential cohomology is in

Last revised on September 7, 2018 at 13:03:38. See the history of this page for a list of all contributions to it.