Todd class

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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 αi as its Chern roots, then

td(E) = Π Q(αi),

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 + c1/2 + (c12+c2)/12 + c1c2/24 + ...

where the cohomology classes ci are the Chern classes of E, and lie in the cohomology group H2i(X). If X is finite dimensional then most terms vanish and td(E) is a polynomial in the Chern classes.

[edit] 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)
  • M.I. Voitsekhovskii (2001), “Todd class”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104