Definition
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 (aka, the splitting principle). For the definition, let
be the formal power series with the property that the coefficient of xn in Q(x)n+1 is 1 (where the Bi are Bernoulli numbers).
If E has the αi as its Chern roots, then
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 + (−c14 + 4c12c2 + c1c3 + 3c22 − c4)/720 + ...
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.
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