# 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 Adams (1960) 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 Maunder (1963). However secondary and higher cohomology operations are rather cumbersome to use, and their study was mostly abandoned when Michael Atiyah pointed out in the 1960s that many of their applications could be proved more easily using generalized cohomology theories.

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

## See also

## References

- Adams, J. F. (1960), "On the non-existence of elements of Hopf invariant one",
*Ann. Math.*, The Annals of Mathematics, Vol. 72, No. 1,**72**(1): 20–104, doi:10.2307/1970147, JSTOR 1970147 - Baues, Hans-Joachim (2006),
*The algebra of secondary cohomology operations*, Progress in Mathematics,**247**, Birkhäuser Verlag, ISBN 978-3-7643-7448-8, MR 2220189 - Harper, John R. (2002),
*Secondary cohomology operations*, Graduate Studies in Mathematics,**49**, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3198-4, MR 1913285 - Maunder, C. R. F. (1963), "Cohomology operations of the Nth kind",
*Proceedings of the London Mathematical Society. Third Series*,**13**: 125–154, doi:10.1112/plms/s3-13.1.125, ISSN 0024-6115, MR 0211398