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:

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.

For ordinary cohomology:

For Adams operations:

  • Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Section 10 of: Algebraic topology from a homotopical viewpoint, Springer (2002)

For Steenrod operations and operations in generalized cohomology (MU and BP):

Discussion of refinement of cohomology operations to differential cohomology is in

Last revised on September 3, 2020 at 03:40:31. See the history of this page for a list of all contributions to it.