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 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).
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
Steenrod’s original colloquium lectures were published as:
Review in:
Textbook accounts include the following.
For ordinary cohomology:
For Adams operations:
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.