Commit 34f4197b authored by Jannes Bantje's avatar Jannes Bantje

add new section placeholder

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...@@ -52,8 +52,13 @@ If the product of the $c_i$ has degree $n= \dim M$ we can take the pairing with ...@@ -52,8 +52,13 @@ If the product of the $c_i$ has degree $n= \dim M$ we can take the pairing with
How does this fit into the abstract definition given above? -- Via the frame bundle and the Borel construction real vector bundles correspond to $\GL_r(\mathbb{R})$-principal bundles. How does this fit into the abstract definition given above? -- Via the frame bundle and the Borel construction real vector bundles correspond to $\GL_r(\mathbb{R})$-principal bundles.
In the case of oriented vector bundles the structure group reduces to $\SL_r(\mathbb{R})$ and one can show, that the Euler class satisfies functoriality, such that \cref{def:char_class} is fulfilled. In the case of oriented vector bundles the structure group reduces to $\SL_r(\mathbb{R})$ and one can show, that the Euler class satisfies functoriality, such that \cref{def:char_class} is fulfilled.
In the special case of $E$ being the tangent bundle of a smooth,compact, oriented, $r$-dimensional manifold $M$ its Euler class is an element of the top dimensional cohomology group. In the special case of $E$ being the tangent bundle of a smooth, compact, oriented, $r$-dimensional manifold $M$ its Euler class is an element of the top dimensional cohomology group.
The corresponding characteristic number agrees with the \Index{Euler characteristic} of that manifold (see \cite[Cor.~11.12]{milnor_stasheff}). The corresponding characteristic number agrees with the \Index{Euler characteristic} of that manifold (see \cite[Cor.~11.12]{milnor_stasheff}).
\end{example} \end{example}
% section the_general_concept (end) % section the_general_concept (end)
\section{Multiplicative Sequences} % (fold)
\label{sec:multiplicative_sequences}
% section multiplicative_sequences (end)
% chapter towards_the_topological_index_characteristic_classes (end) % chapter towards_the_topological_index_characteristic_classes (end)
\ No newline at end of file
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