Special and general types
Paths and cylinders
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).
Steenrod’s original colloquium lectures were published as:
- Norman Steenrod, Cohomology operations, and obstructions to extending continuous functions Advances in Math. 8, 371–416. (1972). (scanned pdf)
Textbook accounts include the following.
Robert Mosher, Martin Tangora, p. 38 of Cohomology Operations and Application in Homotopy Theory, Harper and Row (1968) (pdf)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer (2002)
Peter May, chapter 22, section 5 of A concise course in algebraic topology (pdf)
Revised on November 20, 2015 09:08:58
by Urs Schreiber