Commit dd699b89 authored by Jannes Bantje's avatar Jannes Bantje

rough intro of todd class

parent 0a4844dd
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......@@ -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 $2n$.
Then we write
\[
c(X) = 1 + c_1(X) + \ldots + c_n(X) = (1+ x_1) \cdots (1+ x_n)
\]
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}.
\end{example}
% section index_theory (end)
% chapter elliptic_genera_phd_seminar (end)
\ No newline at end of file
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