In mathematics, a secondary cohomology operation is a functorial correspondence between
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
s. More precisely, it is a natural transformation from the kernel of some primary
cohomology operation In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if ''F'' is a functor defining a cohomology theory, then a coho ...
to the cokernel of another primary operation. They were introduced by in his solution to the
Hopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
__TOC__ Motivation
In 1931 Heinz Hopf used Clifford parallels to construct the '' Hopf map''
:\eta\colon S^3 \to S ...
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
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...
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
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist.
Massey triple product
Le ...
, the
Toda bracket In mathematics, the Toda bracket is an operation on homotopy classes of maps, in particular on homotopy groups of spheres, named after Hiroshi Toda, who defined them and used them to compute homotopy groups of spheres in .
Definition
See or fo ...
, and differentials of
spectral sequences.
See also
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Peterson–Stein formula
References
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*{{Citation , last1=Maunder , first1=C. R. F. , title=Cohomology operations of the Nth kind , doi=10.1112/plms/s3-13.1.125 , mr=0211398 , year=1963 , journal=Proceedings of the London Mathematical Society , series=Third Series , issn=0024-6115 , volume=13 , pages=125–154
Algebraic topology