Commit 160be9f7 authored by Jannes Bantje's avatar Jannes Bantje

add important example

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......@@ -3,7 +3,7 @@
\label{cha:elliptic_genera_phd_seminar}
\section{Introduction -- Julian} % (fold)
\section{Introduction: Genera and Multiplicative Sequences -- Julian} % (fold)
\label{sec:introduction_julian}
The story start by looking at bordism invariants.
......@@ -28,7 +28,9 @@ In studying such invariant, we will
\]
\item and higher versions thereof.\todo{specify}
\end{itemize}
\bigskip
We start by recalling some facts from bordism theory starting with an example of a bordism invariant:
\begin{example}
Let $M$ be a closed, $4n$-dimensional oriented smooth manifold.
The cup product gives rise to a symmetric bilinear form on $H^{2n}(M;\mathbb{Q})$.
......@@ -38,14 +40,12 @@ In studying such invariant, we will
\]
is defined as the number of positive Eigenvalues minus the number of negative ones of this symmetric bilinear form.
As the signature is invariant under oriented cobordism, we get a ring homomorphism $\operatorname{sign} \colon \Omega_*^{\SO} \to \mathbb{Z}$.
This bordism invariant has the nice feature, that if $M$ is closed, spin and $\dim M \equiv 4 \mod 8$ then
\[
\operatorname{sign}(M) \equiv 0 \mod 16
\]
\end{example}
One of the main players of bordism theory are the \Index{Thom spectra}, which arise as the sequence of Thom spaces of the universal bundles over $\B G(n)$.
It is denoted by $MG$ and we have
\[
......@@ -53,7 +53,7 @@ It is denoted by $MG$ and we have
\]
\todo{a reference sounds reasonable}
\begin{theorem}{Thom}
\begin{theorem}{Thom}\label{thm:oriented-bordism_ring}
The oriented bordism ring is given by
\[
\Omega^{\SO}_* \otimes \mathbb{Q} = \mathbb{Q} \benbraceX[\big]{y_{4k} \given k \in \mathbb{N}}
......@@ -61,7 +61,45 @@ It is denoted by $MG$ and we have
where $y_{4k}= \benbrace*{\mathbb{C}\mathbb{P}^{2k}} \in \Omega^{\SO}_{4k}$.
\end{theorem}
Furthermore two manifolds represent the same class precisely if all their Prontryagin numbers and Stiefel-Whitney numbers coincide.\todo{reference}
Furthermore two manifolds represent the same class precisely if all their Prontryagin numbers and Stiefel-Whitney numbers coincide.\todo{reference? -- Julian just mentioned Pontryagin numbers?}
Now we can define, what is meant by \enquote{genus} in this context:
\begin{definition}{\cite[Sec.~1.6]{hirzebruch_modularforms}}
Let $R$ be an integral domain over $\mathbb{Q}$.
Then a \Index{genus} is a unital ring homomorphism
\[
\varphi \colon \Omega^{\SO}_* \otimes \mathbb{Q} \To{} R
\]
\end{definition}
Since we know $\Omega^{\SO}_*$ by \cref{thm:oriented-bordism_ring}, we study these genera by less abstract, more computational means, as we are going to explain now:
The general idea is to consider Chern resp. Pontryagin classes formally as elementary symmetric functions, \cite[Sec.~1.4]{hirzebruch_modularforms}.
\begin{example}
Let $E = E_1 \oplus \ldots \oplus E_n$ be a direct sum of complex line bundles and set $x_i = c_1(E_i) \in H^2(X;\mathbb{Z})$.
By \cref{thm:axioms_chern} we have
\[
c(E) = (1 + x_1) \cdots (1+ x_n) = 1 + c_1 + \ldots + c_r
\]
where the Chern class $c_r= \sigma(x_1, \ldots ,x_n)$ is the $r$-th \Index{elementary symmetric function} in the $x_i$.
By assuming the existence of Hermitian structures on the $E_1$, we can reduce their structure groups from $\mathbb{C}^\times$ to $T^1=S^1\cong\Unitary(1)$ and therefore the one of $E$ to
\[
T^n = \set*{A \in \Unitary(n) \given A = \operatorname{diag} \enbrace*{e^{2 \pi i \varphi_1},\ldots , e^{2 \pi i \varphi_n}}, \varphi_i \in \mathbb{R}}
\]
which is a maximal torus of the connected Lie group $\Unitary(n)$.
All such maximal tori can be conjugated into each other and such a conjugation by, say, $g \in \Unitary(n)$ permutes the $S^1$ factors of $T^n$ but still yields an isomorphism of the $\Unitary(n)$-bundle (but not necessarily of the $T^n$-bundle).
Since an explicit expression of genus has to be invariant under such $\Unitary(n)$-isomorphisms, it must remain fixed under permutations of the $x_i$.
And therefore it has to be a polynomial in the Chern classes (which we identified as elementary symmetric polynomials) by the fundamental theorem on symmetric polynomials!
A similar story can be told for the Pontryagin classes, see \cite[8]{hirzebruch_modularforms}.
\end{example}
\begin{proposition}
There is a one-to-one correspondence of genera $\varphi$ and multiplicative sequences $Q$.
\end{proposition}
% section introduction_by_julian (end)
\section{Modular Forms -- Jens} % (fold)
......
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