Contents

cohomology

# 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.

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

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

If here $E_\bullet$ and $F_\bullet$ 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

### Cup powers in multiplicative cohomology

###### Example

[Cup-square cohomology operation]

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

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

Then for all $n \in \mathbb{N}$ there is an unstable $E$-cohomology operation

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

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

(2)$X \;\;\mapsto\;\; \Big( [X, \Omega^\infty \Sigma^n E] \,=\, {\widetilde E}{}^n(X) \overset{ \;\; \Delta_{{\widetilde E}{}^n(X)} \;\; }{\longrightarrow} {\widetilde E}{}^n(X) \times {\widetilde E}{}^n(X) \overset{ \;\; \cup^E_X \;\; }{\longrightarrow} {\widetilde E}{}^n(X) \,=\, [X, \Omega^\infty \Sigma^{2n} E] \Big)$

Here the first function $\Delta_{{\widetilde E}{}^n(X)}$ is the diagonal on the set underlying the degree=$n$ $E$-cohomology group of $X$, while the second function $\cup^E_X$ 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

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

implies that this natural transformation (2) between the representable functors $[-,\Omega^\infty \Sigma^n E]$ and $[-,\Omega^\infty \Sigma^{2n} E]$ 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 $k$th cup power $(-)^{k_\cup}$ for any $k \in \mathbb{N}$. 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: