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 J. Frank 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).

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.

See also


  • Adams, J. Frank (1960), "On the non-existence of elements of Hopf invariant one", Annals of Mathematics, 72 (1): 20–104, CiteSeerX, 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, doi:10.1090/gsm/049, 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
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