@@ -288,4 +288,24 @@ This seems to be mostly algebra and I'm unsure, whether it is actually relevant

\todo{looked at chapter 3 in the book}

\todo[inline]{maybe not as important for my talk!?}

% section formal_group_laws_leon (end)

\section{Index Theory -- Jannes}% (fold)

\label{sec:index_theory}

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}

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!

\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