Commit 34945305 authored by Jannes Bantje's avatar Jannes Bantje

Merge branch 'characteric-and-genera' into 'master'

Characteric and genera

See merge request !3
parents 462f95cd 52191497
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......@@ -534,6 +534,12 @@ This cover is trivialising since we have $G$-maps $p_i \colon V_i \to G$ by cons
In fact being contractible is not just a property of the total space of universal bundles -- it even characterises them completely: A numerable $G$-principal bundle is universal if and only if its total space is contractible, see \cite[Thm.~14.4.12]{tomDieck_algebraic}.
% section existence_of_universal_bundles (end)
\section{Thom Isomorphism} % (fold)
\label{sec:thom_isomorphism}
\todo{add the statement}
% section thom_isomorphism (end)
% chapter appendix_bundle_theory (end)
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......@@ -232,8 +232,8 @@ The symbol gives raise to an extremly important class of differential operators:
\begin{definition}\label{def:elliptic}
Let $D \in \DiffOp^k(E_0,E_1)$ and $x \in M$.
Then we say, that $D$ is \Index{elliptic at} $x$ if for each $\xi \in \Tang_x^* M$, $\xi \neq 0$ the homomorphism $\sigma_k(D)(\xi) \colon (E_0)_x \to (E_1)_x$ is invertible.
$D$ is \Index{elliptic}, if it is elliptic at each point $x \in M$.
Then we say, that $D$ is \bet{elliptic at} $x$ if for each $\xi \in \Tang_x^* M$, $\xi \neq 0$ the homomorphism $\sigma_k(D)(\xi) \colon (E_0)_x \to (E_1)_x$ is invertible.
$D$ is \Index[differential operator!elliptic]{elliptic}\index{elliptic operator}, if it is elliptic at each point $x \in M$.
\end{definition}
......
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......@@ -890,9 +890,54 @@
}
@book{milnor_stasheff,
author = {Milnor, John W. and Stasheff, James D.},
author = {Milnor, John and Stasheff, James D.},
title = {Characteristic classes},
note = {Annals of Mathematics Studies, No. 76},
publisher = {Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo},
year = {1974},
}
@book{hirzebruch_modularforms,
author = {Hirzebruch, Friedrich and Berger, Thomas and Jung, Rainer},
title = {Manifolds and modular forms},
series = {Aspects of Mathematics, E20},
publisher = {Friedr. Vieweg \& Sohn, Braunschweig},
year = {1992},
doi = {10.1007/978-3-663-14045-0},
}
@book{hirzebruch_Lgenus,
author = {Hirzebruch, F.},
title = {Neue topologische {M}ethoden in der algebraischen {G}eometrie},
series = {Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 9},
publisher = {Springer},
year = {1956},
pages = {viii+165},
}
@article{grothendieck_chern,
author = {Grothendieck, Alexander},
title = {La théorie des classes de {C}hern},
journal = {Bulletin de la Société Mathématique de France},
volume = {86},
year = {1958},
pages = {137--154},
url = {http://www.numdam.org/item?id=BSMF_1958__86__137_0},
}
@article{novikov_complex_bordism,
author = {Novikov, S. P.},
title = {Homotopy properties of {T}hom complexes},
journal = {Mat. Sb. (N.S.)},
volume = {57 (99)},
year = {1962},
pages = {407--442}
}
@article{milnor_complex_bordism,
author = {Milnor, John},
title = {On the cobordism ring {$\Omega^{\ast}$} and a complex analogue. {I}},
journal = {American Journal of Mathematics},
volume = {82},
year = {1960},
pages = {505--521},
doi = {10.2307/2372970},
}
\ No newline at end of file
......@@ -189,6 +189,7 @@
\settoheight{\Fredsize}{$\Fred$}
\DeclareMathOperator{\indeX}{ind}
\DeclareMathOperator{\IndeX}{Ind}
\DeclareMathOperator{\topind}{top-ind}
\newcommand{\kkprod}{\otimes}
\newcommand{\FredRed}{\raisebox{0pt}[\Fredsize][\depth]{$\widetilde{\raisebox{0pt}[5.8pt][\depth]{$\Fred$}}$}}
......@@ -223,6 +224,8 @@
\renewcommand{\longrightarrow}{\grenzw{}}
\DeclareMathOperator{\ev}{ev}
\DeclareMathOperator{\Span}{span} % Span
\newcommand{\cupp}{\smallsmile}
\newcommand{\capp}{\smallfrown}
\newlength{\otsize}
\settoheight{\otsize}{$\otimes$}
......@@ -263,6 +266,16 @@
\DeclareMathOperator{\Sp}{Sp}
\DeclareMathOperator{\SO}{SO}
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\Strng}{String}
\newcommand{\CP}{\mathbb{C}\mathbb{P}}
\newcommand{\HP}{\mathbb{H}\mathbb{P}}
% \newcommand{\M}{\mathrm{M}\mkern-3mu}
\newcommand{\thoM}{M\mkern-3mu}
\DeclareMathOperator{\td}{td}
\DeclareMathOperator{\ch}{ch}
\DeclareMathOperator{\Thom}{Th}
\newcommand{\cpt}{\mathrm{cpt}}
\newcommand{\tmf}{\ensuremath{\mathrm{tmf}}}
\DeclareMathOperator{\Map}{Map}
\newcommand{\sa}{\mathrm{sa}}
\DeclareMathOperator{\spectrum}{spec}
......@@ -273,6 +286,11 @@
\newcommand{\Bop}{\mathcal{B}}
\newcommand{\BH}{\Bop(\Hilb)}
\newcommand{\Ahat}{\ensuremath{\widehat{A}}}
\newcommand{\B}[1]{B{#1}} % classifying spaces
\newcommand{\Inf}{I\mkern-.1mu n\mkern-.5mu f}
\newcommand{\Infhat}{I\mkern-.1mu \widehat{n}\mkern-.5mu f}
\newcommand{\RInf}{\mathcal{R}\Infhat}
......@@ -313,6 +331,11 @@
\newcommand{\gradedMR}{\raisebox{0pt}[\Msize][\depth]{$\widehat{\raisebox{0pt}[5.8pt][\depth]{\ensuremath{\MR}}}$}}
% ======================================================================================
%-- Dirac operators and so on
% ======================================================================================
\newcommand{\Dirac}{\slashed{D}}
% ======================================================================================
%-- algebra stuff
% ======================================================================================
......@@ -376,6 +399,7 @@
\newcommand\SetSymbol[1][]{\nonscript\:#1\vert\allowbreak\nonscript\:\mathopen{}}
\providecommand\given{} % to make it exist
\DeclarePairedDelimiterX\set[1]\{\}{\renewcommand\given{\SetSymbol[\delimsize]}#1}
\DeclarePairedDelimiterX\benbraceX[1]{[}{]}{\renewcommand\given{\SetSymbol[\delimsize]}#1}
% ======================================================================================
%-- definition of mappings
......
......@@ -65,6 +65,7 @@
\usepackage{xfrac}
\usepackage{mathdots} % Verbesserung von Punkten wie zB \ldots
\usepackage{centernot}
\usepackage{slashed}
\usepackage{stackrel}
\usepackage{nicematrix}
\DeclareSymbolFont{bbold}{U}{bbold}{m}{n}
......
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