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)
......@@ -9,6 +9,7 @@ Some of the background needed for this chapter is given in \cref{sec:fibre_bundl
\todo[inline]{weave in some of the theory in \textcite[II.§11, Appendix B]{lawson_spin} (including multiplicative sequences)}
\todo[inline]{mention a few of the highlights of \textcite{milnor_stasheff}}
\todo[inline]{try to \emph{also} incorporate \textcite{hirzebruch_modularforms}}
\section{The General Concept} % (fold)
\label{sec:the_general_concept}
......@@ -16,11 +17,12 @@ Let $G$ be a topological group (Lie groups will be of particular interest).
In the Lie group setting we require all maps to be smooth.
We denote the set of isomorphism classes of $G$-principal bundles over $X$ with $\PB_G(X)$.
Via the pullback $\PB_G(-)$ is a contravariant functor $\PB_G(-) \colon \TOP \to \SET$, which is homotopy invariant, if we require our bundles to be numerable.\footnote{or if we restrict ourselves to paracompact spaces such as manifolds}
Via the pullback of principal bundles $\PB_G(-)$ is a contravariant functor $\PB_G(-) \colon \TOP \to \SET$, which is homotopy invariant, if we require our bundles to be numerable.\footnote{or if we restrict ourselves to paracompact spaces such as manifolds}
Isomorphism classes of $G$-principal bundles are classified via the universal bundle $EG \to BG$, i.e.\ the following map is a bijection
\mapdef{\iota_X \colon \benbrace*{X,BG}}{\PB_G(X)}{\benbrace*{f}}{\benbrace*{f^*EG}}{}
The existence for universal bundles is covered in \cref{sec:existence_of_universal_bundles}.
In this language we can define characteristic classes as follows:
\begin{definition}\label{def:char_class}
A \Index{characteristic class} is a natural transformation of $\PB_G(-)$ to some cohomology functor $h^k(-)$ regarded as a functor to $\SET$.
\end{definition}
......@@ -38,27 +40,123 @@ Obvious cases of interest for $h^k(-)$ are singular cohomology with coefficients
Characteristic classes give raise to characteristic \emph{numbers}: Many cohomology theories have a product structure, for example the cup product of singular cohomology.
Suppose the base space is a manifold $M$ and that we have some characteristic classes $c_1, \ldots, c_k$ living in $H^*(M)$.
If the product of the $c_i$ has degree $n= \dim M$ we can take the pairing with the fundamental class $\benbrace*{M}$, yielding a \Index{characteristic number}.
One should note, that in general there is no single characteristic number associated to some type of characteristic classes, but rather several characteristic numbers, as different partitions of $n$ might be available.
When working with a differentiable manifold $M$ it is also standard to consider the tangent bundle $\Tang M$ and define the characteristic classes and numbers of $M$ as the ones of $\Tang M$.
\subsection*{Euler Classes}
Let $E \to X$ be a real \emph{oriented} vector bundle of rank $r$.
The orientation of $E$ amounts to a continuous choice of generators of the cohomology group $H^r(F,F\setminus F_0;\mathbb{Z}) \cong \mathbb{Z}$ for each fibre $F$ and corresponding zero section $F_0$.
The Thom isomorphism (see \cref{sec: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
\[
H^r(E, E \setminus E_0;\mathbb{Z}) \longrightarrow H^r(X;\mathbb{Z})
\]
The \Index{Euler class} $e(E)$ is the image of $u$ under this morphism.
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.
The corresponding characteristic number agrees with the \Index{Euler characteristic} of that manifold (see \cite[Cor.~11.12]{milnor_stasheff}).
\begin{lemma}
The Euler class has a few nice properties (besides functoriality):
\begin{enumerate}[(i)]
\item Whitney sum formula: $e(E \oplus F) = e(E) \cupp e(F)$
\item Orientation: If $\overline{E}$ denotes the opposite orientation $e(\overline{E}) = - e(E)$
\item Normalisation: If $E$ has a nowhere-zero section, then $e(E)=0$
\end{enumerate}
\end{lemma}
\todo{find some reference}
\subsection*{Stiefel Whitney Classes}
\todo[inline]{Stiefel Whitney class}
\subsection*{Chern Classes}
Chern classes can be approached on an axiomatic level.
The following list is from the book by \textcite[Sec.~1.2]{hirzebruch_modularforms}.
There is another set of axioms by \textcite{grothendieck_chern}, which uses less axioms.
\begin{theorem}\label{thm:axioms_chern}
For each complex vector bundle $E$ over a manifold $X$ there exist \Index{Chern classes} $c_i(E) \in H^{2i}(X;\mathbb{Z})$ and a \Index{total Chern class}
\(
c(E) \coloneqq \sum_{i=0}^{\infty} c_i(E) \in H^*(X;\mathbb{Z})
\)
with
\begin{enumerate}[(i)]
\item $c_0(E) = 1$
\item $c_i(f^* E) = f^* c_i(E)$
\item $c(E \oplus F) = c(E) \cupp c(F)$
\item The Hopf bundle $H$ over $\mathbb{C}P^n$ satisfies
\[
c(H) = 1-g
\]
where $g \in H^2(\mathbb{C}P^n;\mathbb{Z})$ is the generating element of the cohomology ring of $\mathbb{C}P^n$
\end{enumerate}
The Chern classes are uniquely determined by these properties.
\end{theorem}
Chern classes can be constructed in several ways.
\Textcite{milnor_stasheff} use the observation, that any complex bundle comes with a canonical orientation\footnote{which boils down to the fact, that $\GL_n(\mathbb{C})$ is connected, see \cite[Lem.~14.1]{milnor_stasheff}} which allows one to use the top dimensional Euler class as the basic definition and continue inductively from there on.
\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 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
Let $\pi \colon E \to B$ be a complex bundle\footnote{over a paracompact space $B$} and consider $B$ as as subset of $E$ via the zero-section.
We let $E_0 = E \setminus B$ and note, that a point in $E_0$ is given by a fibre $F$ and a non-zero vector $v$ in it.
Now we build a new bundle $\omega_0$ over $E_0$ by defining a fibre over this point to consist of the orthogonal complement of $v$ in the vector space $F$.\footnote{we assume $E$ to be equipped with a Hermitian structure here}
This bundle $\omega_0$ has rank one less than $E \to B$.
Recall that the Gysin sequence associated to the fibre bundle $\pi_0 \colon E_0 \to B$ is given by
\[
\begin{tikzcd}
\ldots \rar & H^{i-2n}(B) \rar["\cupp e"] & H^i(B) \rar["\pi_0^*"] & H^i(E_0) \rar & H^{i-2n+1}(B) \rar & \ldots
\end{tikzcd}
\]
and $H^{i-2n}(B) = H^{i-2n+1}(B) =0$ for $i < 2n-1$, such that $\pi_0^*$ is an isomorphism in this case.
This allows us to define the Chern classes as follows:
\[
c_k(E) \coloneqq \begin{cases}
(\pi_0^*)^{-1} \enbrace[\big]{c_k(\omega_0)} &\text{ if }k < n\\
e(E_\mathbb{R}) &\text{ if } k=n \\
0 &\text{ if } k > n
\end{cases}
\]
Checking the properties above requires some work.
Chern classes can also be defined via Chern classes over the universal bundle, which apparently can be described using Schubert cycles from algebraic geometry.
\todo[inline]{what is Chern-Weil theory? -- differential topology approach}
\subsection*{Pontryagin Classes}
Closely related to Chern classes, there are the Pontryagin classes:
\begin{definition}
Let $E$ be a real vector bundle over $X$.
The \Index{Prontryagin classes} $p_i(E)$ are defined as
\[
H^r(E, E \setminus E_0;\mathbb{Z}) \longrightarrow H^r(X;\mathbb{Z})
p_i(E) \coloneqq (-1)^i \cdot c_{2i}(E \otimes \mathbb{C}) \in H^{4i}(X;\mathbb{Z})
\]
The \Index{Euler class} $e(E)$ is the image of $u$ under this morphism.
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.
The corresponding characteristic number agrees with the \Index{Euler characteristic} of that manifold (see \cite[Cor.~11.12]{milnor_stasheff}).
\end{example}
\end{definition}
The following properties can be derived from \cref{thm:axioms_chern}:
\begin{proposition}\label{prop:axioms-pontryagin}
Let $E$ be a real vector bundle over $X$ of rank $n$.
Then the Pontryagin classes $p_i(E) \in H^{4i}(X;\mathbb{Z})$ and the \Index{total Pontryagin class} $p(E) = \sum_{i=0}^{\infty} p_i(E)$ satisfy the following axioms
\begin{enumerate}[(i)]
\item $p_0(E)=1$,
\item $p_i(f^*E) = f^* p_i(E)$,
\item $p(E \oplus F) = p(E) \cupp p(F)$ modulo $2$-torsion,
\item $p(H_\mathbb{R}) = 1 + g^2$, where $H_\mathbb{R}$ is the underlying real bundle of the Hopf bundle.
\end{enumerate}
\end{proposition}
\subsection*{Todd Classes}
Another type of characteristic class closely related to the Chern classes, are the \Index{Todd classes}.
Their formal definition is best given by resorting to the techniques of multiplicative sequences of \cref{cha:elliptic_genera_phd_seminar}.
% section the_general_concept (end)
\section{Multiplicative Sequences} % (fold)
\label{sec:multiplicative_sequences}
\section{Computations of Characteristic Classes} % (fold)
\label{sec:computations_of_characteristic_classes}
% section multiplicative_sequences (end)
% section computations_of_characteristic_classes (end)
% chapter towards_the_topological_index_characteristic_classes (end)
\ No newline at end of file
......@@ -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}
......
This diff is collapsed.
......@@ -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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