diff --git a/contents/characteristic.tex b/contents/characteristic.tex index f95cad45d4276310ad7907a8ccb8d2549279ca13..e43b29d2b9bb1916534802d9c98c87e8339114b1 100644 --- a/contents/characteristic.tex +++ b/contents/characteristic.tex @@ -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}). \end{example} % 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) \ No newline at end of file