Secondary cohomology operation

In mathematics, a secondary cohomology operation is a functorial correspondence between cohomology groups. More precisely, it is a natural transformation from the kernel of some primary cohomology operation to the cokernel of another primary operation. They were introduced by in his solution to the Hopf invariant problem. Similarly one can define tertiary cohomology operations from the kernel to the cokernel of secondary operations, and continue like this to define higher cohomology operations, as in.

Michael Atiyah pointed out in the 1960s that many of the classical applications could be proved more easily using generalized cohomology theories, such as in his reproof of the Hopf invariant one theorem. Despite this, secondary cohomology operations still see modern usage, for example, in the obstruction theory of commutative ring spectra.

Examples of secondary and higher cohomology operations include the Massey product, the Toda bracket, and differentials of spectral sequences.