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, as need, be truncated to a polynomial of high enough degree.