Commit bb939ea3 authored by Jannes Bantje's avatar Jannes Bantje

resolve todo by adding reference

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......@@ -42,7 +42,7 @@ If the product of the $c_i$ has degree $n= \dim M$ we can take the pairing with
\begin{example}{Euler class}
Let $E \to X$ be a real oriented vector bundle of rank $r$.
The orientation of $E$ amounts to a continuous choice of generators of the cohomology group $H^r(F,F\setminus F_0;\mathbb{Z})$ for each fibre $F$.
The Thom isomorphism now tells us, that these generators are actually restrictions of on element $u \in H^r(E, E \setminus E_0;\mathbb{Z})$, the \Index{orientation class}.
The Thom isomorphism now tells us, that these generators are actually restrictions of an element $u \in H^r(E, E \setminus E_0;\mathbb{Z})$, the \Index{orientation class} or \Index{fundamental class}.
If the base $X$ is embedded into $E$ via the zero-section, the maps $(X,\emptyset) \hookrightarrow (E,\emptyset) \hookrightarrow (E, E \setminus E_0)$ induce a morphism
\[
H^r(E, E \setminus E_0;\mathbb{Z}) \longrightarrow H^r(X;\mathbb{Z})
......@@ -52,9 +52,8 @@ 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.
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 compact oriented $r$-dimensional manifold its Euler class is an element of the top dimensional cohomology group.
Now there is an associated characteristic number described above, which agrees with the \Index{Euler characteristic} of that manifold.
\todo[inline]{find some reference?}
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}).
\end{example}
% section the_general_concept (end)
% chapter towards_the_topological_index_characteristic_classes (end)
\ No newline at end of file
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