Todd class

Todd class

In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can 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 bundle does to a normal bundle. The Todd class plays a fundamental role in generalising the classical Riemann-Roch theorem to higher dimensions, in the Hirzebruch-Riemann-Roch theorem and 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 sum of 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 ring of "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.

References

*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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