 ### rough intro of todd class

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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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