Todd class

The Todd class is a construction of the algebraic topology of characteristic classes. The Todd class of a vector bundle can be explained by the theory of Chern classes and exists where they exist, especially in differential topology, the theory of complex manifolds and algebraic geometry. Roughly speaking, it acts like a reciprocal Chern class or is in relation to her like a normal to an co- normal bundle. The Todd class plays a fundamental role in the generalization of the Riemann-Roch theorem to higher dimensions, in set of Hirzebruch - Riemann -Roch or set of Grothendieck - Hirzebruch - Riemann -Roch.

It is named after the English mathematician John Arthur Todd, who introduced a special case in 1937 in algebraic geometry, before the definition of the Chern classes. The geometric idea is sometimes also called Todd - Eger - class, the general definition in higher dimensions is by Friedrich Hirzebruch ( in his book Topological methods of algebraic geometry).

Definition

To define the Todd class to a complex -dimensional vector bundle on a topological space, it is usually possible to a Whitney sum ( ie direct sum ) of line bundles to restrict using a common method in the theory of characteristic classes, the Chern roots. consider

As the formal power series, wherein the coefficients are the Bernoulli numbers. If that has as Chern roots is

Which is calculated in the cohomology ring of ( or in its completion, if one considers infinite-dimensional manifolds ).

The explicit form of the Todd class as a formal power series in the Chern classes is:

The Kohomologieklassen are the Chern classes of and lie in the cohomology group. If is finite, disappear most Terme and is a polynomial in the Chern classes.