From 34f4197bb4725bc4bf19846ef286a9f20a2100b0 Mon Sep 17 00:00:00 2001 From: Jannes Bantje Date: Mon, 3 Feb 2020 16:11:54 +0100 Subject: [PATCH] add new section placeholder --- contents/characteristic.tex | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/contents/characteristic.tex b/contents/characteristic.tex index f95cad4..e43b29d 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 -- 2.26.2