@@ -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.

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}).