TITLE: Hirzebruch's motivation of the Todd class
QUESTION [61 upvotes]: In Prospects in Mathematics (AM-70), Hirzebruch gives a nice discussion of why the formal power series $f(x) = 1 + b_1 x + b_2 x^2 + \dots$ defining the Todd class must be what it is. In particular, the key relation $f(x)$ must satisfy is that 
($\star$) the coefficient of $x^n$ in $(f(x))^{n+1}$ is 1 for all $n$.
As Hirzebruch observes, there is only one power series with constant term 1 satisfying that requirement, namely
$$f(x) = \frac{x}{1-e^{-x}} = 1 + \frac{x}{2}+\sum_{k\geq 2}{B_{k}\frac{x^{k}}{k!}} = 1 + \frac{x}{2} + \frac{1}{6}\frac{x^2}{2} - \frac{1}{30}\frac{x^4}{24} + \dots,$$
where the $B_k$ are the Bernoulli numbers. 
The only approach I see to reach this conclusion is: 

Use ($\star$) to find the first several terms: $b_1 = 1/2, b_2 = 1/12, b_3 = 0, b_4 = -1/720$.
Notice that they look suspiciously like the coefficients in the exponential generating function for the Bernoulli numbers, so guess that $f(x) = \frac{x}{1-e^{-x}}$.
Do a residue calculation to check that this guess does satisfy ($\star$).


My question is whether anyone knows of a less guess-and-check way to deduce from ($\star$) that $f(x) = \frac{x}{1-e^{-x}}$.

REPLY [5 votes]: A pure topological derivation of the characteristic power series of the Todd class can be obtained by looking at pushforward maps in complex oriented cohomology theories. This is an outline of the argument: basically, since both topological complex $K$-theory and even $2$-periodic rational singular cohomology are complex oriented cohomology theories, it is possible to define integration maps (for stably complex $X$):
$$
\int_X^{K}:K(X) \to K(\mathrm{pt}) \cong \mathbb{Z}
$$
$$
\int_X^{HP_{\mathrm{ev}}\mathbb{Q}}: \prod_{i\in \mathbb{Z}}H^{2i}(X;\mathbb{Q}) \to \prod_{i\in \mathbb{Z}}H^{2i}(\mathrm{pt};\mathbb{Q}) \cong \mathbb{Q}.
$$
It is easy to see, by doing some consideration on how integration maps are defined, that these integration maps are non natural with respect to the Chern character $\mathrm{ch}$, and that there exists a class $\mathrm{td}_X \in \prod_{i\in \mathbb{Z}}H^{2i}(X;\mathbb{Q})$ (which is defined to be the Todd class of $X$) such that
$\require{AMScd}$
\begin{CD}
K(X) @>{\mathrm{ch}_X(-)\cdot \mathrm{td}_X}>> \prod_{i \in \mathbb{Z}}H^{2i}(X;\mathbb Q)\\
@V{\int_X^K}VV @VV{\int_X^{HP_{\mathrm{ev}}\mathbb{Q}}}V \\
\mathbb{Z} @>{\mathrm{ch}_{\mathrm{pt}}}>> \mathbb{Q}
\end{CD}
is commutative. The integration maps are defined by means of the Thom isomorphisms (multiplications by the Thom class) for complex vector bundles in a complex oriented cohomology theory. Now, by using the properties of these Thom classes and an application of the splitting principle, it is possible to show that the Todd class $\mathrm{td}_X$ can be expressed as a product of pullbacks of single cohomology class, namely
$$
\dfrac{c_1(\mathcal{O}(1))}{1-e^{-c_1(\mathcal{O}(1))}} \in \prod_{i \in \mathbb{Z}}H^{2i}(\mathbb{P}^\infty;\mathbb{Q}) \cong \mathbb{Q}[[c_1(\mathcal{O}(1))]],
$$
which shows that the characteristic power series of the Todd class is exactly $f(t)=t/(1-e^{-t})$.