@@ -297,16 +297,29 @@ Before we can get started, we need yet another genus to get things going.

In order to define this properly we need to talk about \emph{complex} genera: in \cref{sec:introduction_julian} we used the real cobordism ring $\Omega^{\SO}_*$ as the domain for genera.

Defining an analogue for complex manifolds is a little more complicated, since a cobordism between complex manifolds cannot admit a complex structure for dimension reasons.

This can however be done for \Index[stably almost complex manifold]{stably almost complex manifolds}.

\Textcite{novikov_complex_bordism,milnor_complex_bordism} proved results similar to the ones in \cref{sec:introduction_julian} for the complex bordism ring.\footnote{see \url{http://www.map.mpim-bonn.mpg.de/Complex_bordism} for a nice overview}

\Textcite{novikov_complex_bordism,milnor_complex_bordism} proved results similar to the ones in \cref{sec:introduction_julian} for the complex bordism ring$\Omega^{\Unitary}$.\footnote{see \url{http://www.map.mpim-bonn.mpg.de/Complex_bordism} for a nice overview}

It turns out, that there is an analogous one-to-one correspondence between complex genera and power series $Q(x)$, but these need no longer be even!

Nonetheless, on can proceed in the same fashion: let $X$ be a compact, almost complex manifold of real dimension $2n$.

and get a genus out of $Q$ by setting $\varphi(X)=\enbrace[\big]{\prod_{i=1}^n Q(x_i)}[X]$.

From a computational point of view, one could say, that we now plug the Chern classes instead of the Pontryagin classes into a multiplicative sequence.

\todo[inline]{magic numbers $s_n$}

\begin{example}

We consider the formal power series

\[

Q(x)=\frac{x}{1- e^{-x}}=1+\frac{1}{2} x +\frac{1}{12} x^2+\ldots

\]

We have $f(x)=1-e^{-x}$ in this case.

The associated genus is called \Index{Todd genus}.

This also gives the notion of the \Index{Todd class} and the associated multiplicative sequence is sometimes called \Index{Todd sequence}.