\section{Introduction: Genera and Multiplicative Sequences -- Julian}% (fold)

\label{sec:introduction_julian}

The story start by looking at bordism invariants.

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@@ -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 $H^{2n}(M;\mathbb{Q})$.

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@@ -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 $\dim M \equiv4\mod8$ then

\[

\operatorname{sign}(M)\equiv0\mod16

\]

\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

\[

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@@ -53,7 +53,7 @@ It is denoted by $MG$ and we have

\Omega^{\SO}_*\otimes\mathbb{Q}=\mathbb{Q}\benbraceX[\big]{y_{4k}\given k \in\mathbb{N}}

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@@ -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:

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 $Q$.