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| algebraic topology | |
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: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 Bundle s 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 Polynomial s ''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 Polynomial s 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 . |
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