group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
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.
(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.
every universal characteristic class is a cohomology operation.
Steenrod squares are the stable cohomology endo-operations on ordinary cohomology (mod 2)
Adams operations are the endo-cohomology operations on K-theory
[Cup-square cohomology operation]
Let $E$ be a multiplicative Whitehead-generalized cohomology theory represented by a ring spectrum
Then for all $n \in \mathbb{N}$ there is an unstable $E$-cohomology operation
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 $n$:
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
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).
Steenrod’s original colloquium lectures:
Background and review:
John Michael Boardman, Stable 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)
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)
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)
Operations on topological K-theory:
For Adams operations:
More:
For Steenrod operations on complex-oriented generalized cohomology (such as MU and BP):
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.