- Todd class
mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topologyof characteristic classes. The Todd class of a vector bundlecan be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably in differential topology, the theory of complex manifolds and algebraic geometry. In rough terms, a Todd class acts like a reciprocal of a Chern class, or stands in relation to it as a conormal bundledoes to a normal bundle. The Todd class plays a fundamental role in generalising the classical Riemann-Roch theoremto higher dimensions, in the Hirzebruch-Riemann-Roch theoremand Grothendieck-Hirzebruch-Riemann-Roch theorem.
It is named for
J. A. Todd, who introduced a special case of the concept in algebraic geometry in 1937, before the Chern classes were defined. The geometric idea involved is sometimes called the Todd-Eger class. The general definition in higher dimensions is due to Hirzebruch.
To define the Todd class "td"("E") where "E" is a complex vector bundle on a
topological space"X", it is usually possible to limit the definition to the case of a Whitney sumof line bundles, by means of a general device of characteristic class theory, the use of Chern roots. For the definition, let
:"Q"("x") = "x"/(1 − "e"−"x")
considered as a
formal power series; the expansion can be made explicit in terms of Bernoulli numbers. If "E" has the α"i" as its Chern roots, then
:"td"("E") = Π "Q"(α"i"),
which is to be computed in the
cohomology ringof "X" (or in its completion if one wants to consider infinite dimensional manifolds).
The Todd class can be given explicitly as a formal power series in the Chern classes as follows:
:"td"("E") = 1 + "c"1/2 + ("c"12+"c"2)/12 + "c"1"c"2/24 + ...where the cohomology classes "c""i" are the Chern classes of "E", and lie in the cohomology group H2"i"("X"). If "X" is finite dimensional then most terms vanish and "td"("E") is a polynomial in the Chern classes.
*J. Todd, "The arithmetical theory of algebraic loci", Proc. London Math. Soc., 43 (1937) pp. 190–225
*F. Hirzebruch, "Topological methods in algebraic geometry", Springer (1978)
*springer|id=T/t092930|title=Todd class|author=M.I. Voitsekhovskii
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