Commit 04dfa4fa by Jannes Bantje

### start work on complex genera

parent 6cc928f0
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 ... ... @@ -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 $Q(x) = \frac{x}{1- e^{-x}} = 1 + \frac{1}{2} x + \frac{1}{12} x^2 + \ldots$ \end{example} % section index_theory (end) % chapter elliptic_genera_phd_seminar (end) \ No newline at end of file
 ... ... @@ -922,4 +922,22 @@ year = {1958}, pages = {137--154}, url = {http://www.numdam.org/item?id=BSMF_1958__86__137_0}, } @article{novikov_complex_bordism, author = {Novikov, S. P.}, title = {Homotopy properties of {T}hom complexes}, journal = {Mat. Sb. (N.S.)}, volume = {57 (99)}, year = {1962}, pages = {407--442} } @article{milnor_complex_bordism, author = {Milnor, J.}, title = {On the cobordism ring {$\Omega^{\ast}$} and a complex analogue. {I}}, journal = {American Journal of Mathematics}, volume = {82}, year = {1960}, pages = {505--521}, doi = {10.2307/2372970}, } \ No newline at end of file
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