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 morphisms 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 Whitehead-generalized cohomology theories) then, by the Yoneda lemma, (non-abelian/unstable) cohomology operations are in natural bijection with homotopy classes of maps between classifying spaces.

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

(This statement is made fully explicit for instance below def. 12.3.22 in (Aguilar-Gitler-Prieto).

If here E E_\bullet and F F_\bullet are the component spaces of spectra and if such cohomology operations are compatible with the suspension isomorphism, then one speaks of a stable cohomology operation.

Examples

List of examples

Cup powers in multiplicative cohomology

Example

[Cup-square cohomology operation]

Let EE be a multiplicative Whitehead-generalized cohomology theory represented by a ring spectrum

(E,1 E,m E)CommutativeMonoids(Ho(Spectra),𝕊,). \big( E, 1^E, m^E \big) \;\in\; CommutativeMonoids \Big( Ho\big( Spectra\big), \mathbb{S}, \wedge \Big) \,.

Then for all nn \in \mathbb{N} there is an unstable EE-cohomology operation

(1)[Ω Σ nE() 2 Ω Σ 2nE]E˜ 2n(Ω nE) \big[ \Omega^\infty \Sigma^n E \overset{ (-)^{2_\cup} }{\longrightarrow} \Omega^\infty \Sigma^{2 n} E \big] \;\;\in\;\; \widetilde E^{ 2 n } \big( \Omega^{\infty - n} E \big)

defined – via the Yoneda lemma on the opposite of the classical homotopy category of pointed topological spaces – by acting over any XHo(PointedTopologicalSpaces)X \,\in\, Ho\big(PointedTopologicalSpaces\big) as the cup square on E-cohomology on cohomology in degree nn:

(2)X([X,Ω Σ nE]=E˜ n(X)Δ E˜ n(X)E˜ n(X)×E˜ n(X) X EE˜ n(X)=[X,Ω Σ 2nE]) X \;\;\mapsto\;\; \Big( [X, \Omega^\infty \Sigma^n E] \,=\, {\widetilde E}{}^n(X) \overset{ \;\; \Delta_{{\widetilde E}{}^n(X)} \;\; }{\longrightarrow} {\widetilde E}{}^n(X) \times {\widetilde E}{}^n(X) \overset{ \;\; \cup^E_X \;\; }{\longrightarrow} {\widetilde E}{}^n(X) \,=\, [X, \Omega^\infty \Sigma^{2n} E] \Big)

Here the first function Δ E˜ n(X)\Delta_{{\widetilde E}{}^n(X)} is the diagonal on the set underlying the degree=nn EE-cohomology group of XX, while the second function X E\cup^E_X is the operation of forming the E-cup product of pairs of its elements.

Both of these operations are clearly natural transformations (for the first this is evident, for the second this comes down to the fact that the smash-monoidal diagonals of suspension spectra are natural, hence are indeed monoidal diagonals). Therefore the fully-faithfulness of the Yoneda embedding

Ho(PointedTopologicalSpaces)A[,A]Functors(Ho(PointedTopologicalSpaces) op,Set) Ho \big( PointedTopologicalSpaces \big) \;\overset{ \;\; A \mapsto [-,A]\;\; }{\hookrightarrow}\; Functors \Big( Ho \big( PointedTopologicalSpaces \big)^{op} , Set \Big)

implies that this natural transformation (2) between the representable functors [,Ω Σ nE][-,\Omega^\infty \Sigma^n E] and [,Ω Σ 2nE][-,\Omega^\infty \Sigma^{2n} E] is itself represented by a morphism (1) between the representing classifying spaces, hence by an unstable cohomology operation.

The analogous construction of course exists for the kkth cup power () k (-)^{k_\cup} for any kk \in \mathbb{N}. Brief mentioning of this is in Wirthmüller 12, Example (1) on p. 44 (46 of 67).

References

General

Steenrod’s original colloquium lectures:

Background and review:

On the structure on the collection of all unstable cohomology operations:

On ordinary cohomology

Operations on ordinary cohomology:

On topological K-theory

Operations on topological K-theory:

For Adams operations:

More:

On complex-oriented cohomology theories

For Steenrod operations on complex-oriented generalized cohomology (such as MU and BP):

On differential cohomology

Refinement to differential cohomology:

Last revised on January 25, 2021 at 10:33:37. See the history of this page for a list of all contributions to it.