@@ -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}).