Adams operation

In mathematics, an Adams operation

:ψ^"k"

is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams. The basic idea is to implement some fundamental identities in symmetric function theory, at the level of vector bundles or other representing object in more abstract theories. Here "k" ≥ 0 is a given integer.

The fundamental idea is that for a vector bundle "V" on a topological space "X", we should have

:ψ^"k"("V") is to Λ^"k"("V")

as

:the power sum Σ α^"k" is to the "k"-th elementary symmetric function σ_"k"

of the roots α of a polynomial "P"("t"). (Cf. Newton's identities.) Here Λ^"k" denotes the "k"-th exterior power. From classical algebra it is known that the power sums are certain integral polynomials "Q"_"k" in the σ_"k". The idea is to apply the same polynomials to the Λ^"k"("V"), taking the place of σ_"k". This calculation can be defined in a "K"-group, in which vector bundles may be formally combined by addition, subtraction and multiplication (tensor product). The polynomials here are called Newton polynomials (not, however, the Newton polynomials of interpolation theory).

Justification of the expected properties comes from the line bundle case, where "V" is a Whitney sum of line bundles. For that case treating the line bundle direct factors formally as roots is something rather standard in algebraic topology (cf. the Leray-Hirsch theorem). In general a mechanism for reducing to that case comes from the splitting principle for vector bundles.

References

* Adams, J.F. "Vector Fields on Spheres", The Annals of Mathematics, 2nd Ser., Vol. 75, No. 3 (May, 1962), pp. 603-632