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-Riemann-Roch theorem.

It is named for J. A. Todd, who introduced the concept in algebraic geometry in the 1930s; before the Chern classes were defined, that is. The geometric idea involved is sometimes called the Todd-Eger class. The contemporary definition therefore reverses the history.

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. The essential cases are such that the Chern roots are nilpotent elements of the cohomology ring, so that this definition is unproblematic, and Q could, at need, be truncated to a polynomial of high enough degree.