 ### correspondence genera and multiplicative seq

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 ... ... @@ -139,7 +139,7 @@ Closely related to Chern classes, there are the Pontryagin classes: The following properties can be derived from \cref{thm:axioms_chern}: \begin{proposition} \begin{proposition}\label{prop:axioms-pontryagin} Let $E$ be a real vector bundle over $X$ of rank $n$. Then the Pontryagin classes $p_i(E) \in H^{4i}(X;\mathbb{Z})$ and the \Index{total Pontryagin class} $p(E) = \sum_{i=0}^{\infty} p_i(E)$ satisfy the following axioms \begin{enumerate}[(i)] ... ...
 ... ... @@ -96,12 +96,66 @@ The general idea is to consider Chern resp. Pontryagin classes formally as eleme A similar story can be told for the Pontryagin classes, see \cite{hirzebruch_modularforms}. \end{example} \Textcite[Sec.~1.5]{hirzebruch_modularforms} continue to define the \Index{Chern character} and prove general formulas, that can be applied to $\Lambda^k E$ or $E^*$ for example. Let us systematically construct genera: We consider an even power series $Q(x) = 1 + a_2 x^2 + a_4x^4+ \ldots$ with coefficients in $R$. If we declare the weight of some indeterminates $x_i$ to be two, the product $Q(x_1) \cdots Q(x_n) = 1 + a_2 \sum_{i=1}^n x_i^2 + \ldots$ is symmetric in the $x_i^2$. Once again this translates into the term of weight $4r$ being a homogeneous polynomial $K_r(p_1, \ldots ,p_r)$ of weight $4r$ in the elementary symmetric functions $p_j$ of the $x_i^2$, i.e. $Q(x_1) \cdots Q(x_n) = 1+ K_1(p_1) + K_2(p_1,p_2) + \ldots + K_n(p_1,\ldots ,p_n) + K_{n+1}(p_1,\ldots ,p_n,0) + \ldots$ The sequence of polynomial $(K_r)_r$ is called the \Index{multiplicative sequence} associated to $Q$. This can be used to define a genus: \begin{definition} The genus $\varphi_Q$ is defined for every compact, oriented, smooth manifold $M$ of dimension $4n$ by $\varphi_Q(M) \coloneqq K_n(p_1, \ldots ,p_n)[M] \in \mathbb{R}$ with $p_i = p_i(M) \in H^{4i}(M;\mathbb{Z})$. We put $\varphi_Q(M) \coloneqq 0$ if $4 \nmid n$. With $K(\Tang M) \coloneqq K(p_1, \ldots ,p_n) \coloneqq 1 + K_1(p_1) + K_2(p_1,p_2) + \ldots$ and $K(M) \coloneqq K(\Tang M)[M]$ we have $\varphi_Q(M) = K(M)$. \end{definition} One obviously has to check, that this is in fact a genus, see \cite{hirzebruch_modularforms}. One can now define $f(x) \coloneqq x/Q(x)$ and obtains an odd power series with coefficients in $R$. Let $g$ be the (formal) inverse of $f$, called the \Index{logarithm of the genus $\varphi_Q$}. By resorting to the total Chern and Pontryagin classes of $\mathbb{C}\mathbb{P}^n$ in \cref{thm:axioms_chern,prop:axioms-pontryagin} and a tricky computation, one can show, that $g'(y) = \sum_{n=0}^{\infty} \varphi_Q \enbrace*{\mathbb{C}\mathbb{P}^n} \cdot y^n$ that is, its derivative essentially determined by the values of $\varphi_Q$ at the complex projective spaces, see \cite{hirzebruch_modularforms}. But this also canonically associates a genus $\varphi$ with an even power series by looking at its values on $\CP^n$. \begin{proposition} There is a one-to-one correspondence of genera $\varphi$ and multiplicative sequences $Q$. \end{proposition} We shall not distinguish between a genus and the corresponding multiplicative sequence and make the following convention: $\varphi(M)$ denotes the value of the genus $\varphi$ on the manifold $M$ and for a bundle $E$ over $M$, $\varphi(E)$ denotes the corresponding expression in the Pontryagin classes of $E$. We therefore have $\varphi(M) = \varphi(\Tang M)[M]$ % section introduction_by_julian (end) \section{Ellipctic Genera -- Markus} % (fold) \label{sec:ellipctic_genera_markus} % section ellipctic_genera_markus (end) \section{Modular Forms -- Jens} % (fold) \label{sec:modular_forms_jens} ... ...
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