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.
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
April 13, 2024
secondary, cohomology, operation, this, article, includes, list, references, related, reading, external, links, sources, remain, unclear, because, lacks, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, october, 2. This article includes a list of references related reading or external links but its sources remain unclear because it lacks inline citations Please help improve this article by introducing more precise citations October 2023 Learn how and when to remove this template message 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 editPeterson Stein formulaReferences editAdams J Frank 1960 On the non existence of elements of Hopf invariant one Annals of Mathematics 72 1 20 104 CiteSeerX 10 1 1 299 4490 doi 10 2307 1970147 JSTOR 1970147 Baues Hans Joachim 2006 The algebra of secondary cohomology operations Progress in Mathematics vol 247 Birkhauser Verlag ISBN 978 3 7643 7448 8 MR 2220189 Harper John R 2002 Secondary cohomology operations Graduate Studies in Mathematics vol 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 Retrieved from https en wikipedia org w index php title Secondary cohomology operation amp oldid 1182786782, wikipedia, wiki, book, books, library,
