Hirzebruch's motivation of the Todd class

Question

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:

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

Answer 1

Since you mention playing around with residues, I'm probably not telling you anything you don't already know. But there is a systematic way to extract the power series $f$ from the coefficients of $x^{n-1}$ in $f(x)^{n}$, which goes by the name of the Lagrange inversion formula.

Assume that the constant term of $f$ is invertible, and define $g(x) = \frac{x}{f(x)}$. Then $g(x)$ is a power series which has a compositional inverse. Denote this inverse by $h$, so that if $y = g(x)$ then $x = h(y)$. Write $h(y) = c_1 y + c_2 y^2 + c_3 y^3 + \cdots$. For every integer $n$, the product $n c_n$ is the residue of the differential $\frac{1}{y^n} h'(y) dy = \frac{1}{g(x)^{n}} dx = \frac{ f(x)^{n} }{x^n} dx$, which is the coefficient of $x^{n-1}$ in $f(x)^{n}$.

In your example, you get $c_n = \frac{1}{n}$, so that $h(y) = y + \frac{y^2}{2} + \frac{y^3}{3} + \cdots = - \log(1-y)$. Then $g(x) = 1 - e^{-x}$, so that $f(x) = \frac{x}{1-e^{-x}}$.

Answer 2

The following is basically taking a standard proof of Lagrange inversion and specializing it to your case, but it might amuse. You can rewrite $(\star)$ as $$\frac{1}{2 \pi i} \oint \left(\frac{f(x)}{x}\right)^{n+1} dx =1$$ for all $n$, where the contour surrounds $0$ and is small enough to avoid all other poles of $f$. Multiplying by $y^{n+1}$ and summing on $n$, $$\frac{1}{2 \pi i} \oint \sum_{n=0}^{\infty} \left(\frac{y f(x)}{x}\right)^{n+1} dx =\frac{y}{1-y}$$ or $$\frac{1}{2 \pi i} \oint \frac{dx}{1-y f(x)/x} = \frac{y}{1-y}.$$ Set $g(x)=x/f(x)$. By the holomorphic inverse function theorem, $g$ is invertible near zero, set $h=g^{-1}$.

The only pole of the integrand near $0$ is at $x=h(y)$. The residue at that pole is $$\frac{-1}{y \frac{d}{du} g(u)^{-1}} = \frac{1}{y g(x)^{-2} g'(x)} = \frac{h'(y)}{y y^{-2}} = y h'(y).$$

So $y h'(y) = \frac{y}{1-y}$, $h'(y) = \frac{1}{1-y}$, $h(y) = -\log(1-y)$ (no constant of integration since $h(0)=0$), $g(x) = 1-e^{-x}$ and $f(x) = x/(1-e^{-x})$.

Answer 3

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 $f(t)=t/(1-e^{-t})$.

Answer 4

Hirzebruch was big on the use of number theory and special functions in topology (publishing even in his eighties on Eulerian polynomials and algebraic topology) and was familiar with Norlund's work (Vorlesungen uber Differenzenrechnung, 1924)--citing it in his book--containing a formula for the falling factorial in terms of the generalized Bernoulli polynomials (which appear to be a perennially popular topic in certain math communities):

(in umbral notation here)

$$ (x-1) \cdots (x-n) = (x+\hat{B}.)^n=\hat{B}_n^{(n)}(x)$$

where $$ e^{\hat{B}^{(n)}.(x)t}= ( \frac{t}{e^t-1})^{n+1} e^{xt} .$$

Clearly, $$D^n_{t=0}(\frac{t}{e^t-1})^{n+1}=\hat{B}^{(n)}_n(0)=(-1)^n n! \; ,$$ which is what he required.

(Given the relations between the elementary symmetric functions, the falling factorial, the symmetric power functions and the exterior algebra and characteristic classes, it would be a natural formula for him to note and retain in memory. In fact, the number of $m$-dimensional faces of the $n$-dimensional simplex (with $n+1$ vertices) is $\binom{n+1}{m+1} = \frac{\hat{B}_{m+1}^{(m+1)}(n+2)}{(m+1)!}$ and the number of k-letter words defined by permuting letters attached to the vertices (A068424) is $\frac{(n+1)!}{(n+1-k)!} = \hat{B}_k^{k}(n+2)$.)