- Todd class
In
mathematics , the Todd class is a certain construction now considered a part of the theory inalgebraic topology ofcharacteristic class es. The Todd class of avector bundle can be defined by means of the theory ofChern class es, and is encountered where Chern classes exist — most notably indifferential topology , the theory ofcomplex manifold s andalgebraic geometry . In rough terms, a Todd class acts like a reciprocal of a Chern class, or stands in relation to it as aconormal bundle does to anormal bundle . The Todd class plays a fundamental role in generalising the classicalRiemann-Roch theorem to higher dimensions, in theHirzebruch-Riemann-Roch theorem andGrothendieck-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 aWhitney sum ofline bundle s, by means of a general device of characteristic class theory, the use ofChern 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 ofBernoulli number s. 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
Wikimedia Foundation. 2010.
