nLab cohomology operation

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

List of examples

• every universal characteristic class is a cohomology operation.

• Bockstein homomorphism

• cup product

• Massey product

• power operation

  • Steenrod squares are the stable cohomology endo-operations on ordinary cohomology (mod 2)

  • Adams operations are the endo-cohomology operations on K-theory

• logarithmic cohomology operation

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:

• Norman Steenrod, Cohomology operations, and obstructions to extending continuous functions, Advances in Math. 8, 371–416. (1972) (doi:10.1016/0001-8708(72)90004-7, pdf)

Background and review:

• John Michael Boardman, Stable Operations in Generalized Cohomology [pdf, pdf] in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

• Peter May, chapter 22, section 5 of: A concise course in algebraic topology, University of Chicago Press 1999 (ISBN:978-0226511832, pdf)

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

• John Michael Boardman, David Copeland Johnson, W. Stephen Wilson, Unstable Operations in Generalized Cohomology (pdf), in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

• Andrew Stacey, Sarah Whitehouse, The Hunting of the Hopf Ring, Homology Homotopy Appl. Volume 11, Number 2 (2009), 75-132. (arXiv:0711.3722, euclid:hha/1251832594)

• Tilman Bauer, Formal plethories, Advances in Mathematics Volume 254, 20 March 2014, Pages 497-569 (arXiv:1107.5745, doi:10.1016/j.aim.2013.12.023)

• William Mycroft, Unstable Cohomology Operations: Computational Aspects of Plethories, 2017 (pdf, pdf)

On ordinary cohomology

Operations on ordinary cohomology:

• Robert Mosher, Martin Tangora, Cohomology Operations and Application in Homotopy Theory, Harper and Row (1968), Dover (2008) (ISBN10:0486466647, pdf)

• Andrew Stacey, Sarah Whitehouse, Stable and unstable operations in mod $p$ cohomology theories, Algebr. Geom. Topol. Volume 8, Number 2 (2008), 1059-1091 (arXiv:math/0605471, euclid:agt/1513796856)

On topological K-theory

Operations on topological K-theory:

• Klaus Wirthmüller, Sections 11 and 13 of: Vector bundles and K-theory, 2012 (pdf)

For Adams operations:

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Section 10 of: Algebraic topology from a homotopical viewpoint, Springer (2002) (doi:10.1007/b97586)

More:

• William Mycroft, Sarah Whitehouse, The plethory of operations in complex topological K-theory (arXiv:2001.01608)

On complex-oriented cohomology theories

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

• Stanley Kochman, section 2.5 and 3.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

On differential cohomology

Refinement to differential cohomology:

• Daniel Grady, Hisham Sati, Primary operations in differential cohomology, Adv. Math. 335 (2018), 519-562 (arXiv:1604.05988, doi:10.1016/j.aim.2018.07.019)