Commit 29f4d18a authored by Jannes Bantje's avatar Jannes Bantje

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[8]{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[14]{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[15]{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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