42.65 Todd classes

A final class associated to a vector bundle $\mathcal{E}$ of rank $r$ is its Todd class $Todd(\mathcal{E})$. In terms of the Chern roots $x_1, \ldots , x_ r$ it is defined as

$Todd(\mathcal{E}) = \prod \nolimits _{i = 1}^ r \frac{x_ i}{1 - e^{-x_ i}}$

In terms of the Chern classes $c_ i = c_ i(\mathcal{E})$ we have

$Todd(\mathcal{E}) = 1 + \frac{1}{2}c_1 + \frac{1}{12}(c_1^2 + c_2) + \frac{1}{24}c_1c_2 + \frac{1}{720}(-c_1^4 + 4c_1^2c_2 + 3c_2^2 + c_1c_3 - c_4) + \ldots$

We have made the appropriate remarks about denominators in the previous section. It is the case that given an exact sequence

$0 \to {\mathcal E}_1 \to {\mathcal E} \to {\mathcal E}_2 \to 0$

we have

$Todd({\mathcal E}) = Todd({\mathcal E}_1) Todd({\mathcal E}_2).$

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