Master thesis on Rational Homotopy Theory https://github.com/Jaxan/Rational-Homotopy-Theory
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\documentclass[14pt]{beamer}
\definecolor{todocolor}{rgb}{1, 0.3, 0.2}
\newcommand{\td}[1]{\colorbox{todocolor}{*\footnote{TODO: #1}}}
\input{preamble}
\usepackage{tabularx}
\renewcommand{\tabularxcolumn}[1]{p{#1}}
\graphicspath{ {../presentation/images/} }
\newcommand{\Frame}[2]{
\begin{frame}{#1}#2\end{frame}
}
\title{Rational Homotopy Theory}
\author{Joshua Moerman}
\institute[Radboud Universiteit Nijmegen]{Supervisor: Ieke Moerdijk}
\date{}
\begin{document}
\AtBeginSection[]{
\begin{frame}<beamer>
\tableofcontents[currentsection]
\end{frame}
}
\Frame{}{
\titlepage
}
\section{Introduction to homotopy theory}
\Frame{Homotopy theory}{
\begin{center}
Study of spaces or shapes \\
with ``weak equivalences''
\bigskip
\includegraphics{weak_eqs2}
\end{center}
}
\Frame{Important spaces}{
\begin{align*}
S^1 &= \raisebox{-0.4\height}{\includegraphics{spheres1}} \\[1em]
S^2 &= \raisebox{-0.4\height}{\includegraphics{spheres2}} \\[1em]
S^3 &= \>\> \cdots \\[1em]
&\>\> \vdots
\end{align*}
}
\Frame{Important tool}{
Fundamental group:
\[ \pi_1(X) = \text{maps } S^1 \to X \text{ up to homotopy} \]
\bigskip
\includegraphics{fundamental_group}
}
\Frame{Important tools}{
Homotopy groups:
\begin{align*}
\pi_1(X) &= \text{maps } S^1 \to X \text{ up to homotopy} \\[1em]
\pi_2(X) &= \text{maps } S^2 \to X \text{ up to homotopy} \\[1em]
\pi_3(X) &= \text{maps } S^3 \to X \text{ up to homotopy} \\[1em]
&\>\>\vdots
\end{align*}
}
\Frame{Torsion-free}{
Serre proved in 1950s:
\begin{align*}
\text{odd } k: \quad \pi_n(S^k) \tensor \Q &=
\begin{cases}
\Q &\text{ if } n = k \\
0 &\text{ otherwise }
\end{cases} \\[1em]
\text{even } k: \quad \pi_n(S^k) \tensor \Q &=
\begin{cases}
\Q &\text{ if } n = k, 2k-1 \\
0 &\text{ otherwise }
\end{cases} \\
\end{align*}
}
\section{Rational homotopy theory}
\Frame{Rational homotopy theory}{
\begin{center}
Study of spaces\\
with ``rational equivalences'' \\
and ``rational homotopy groups''
\pause
\bigskip
or
\bigskip
Study of \emph{rational} spaces \\
with weak equivalences \\
and ordinary homotopy groups
\end{center}
}
\Frame{Rational spaces}{
$X$ is \emph{rational} if $\pi_n(X)$ is a $\Q$-vector space
\bigskip
\[ S^1_\Q = \raisebox{-0.55\height}{\includegraphics{infinite_telescope}} \]
}
\section{The main equivalence}
\Frame{Main equivalence}{
\begin{theorem}
\begin{center}
Homotopy theory of rational spaces \\
= \\
Homotopy theory of commutative differential graded algebras
\end{center}
\end{theorem}
}
\Frame{Main equivalence (precise version)}{
\begin{theorem}
\[ \Ho(\Top_{\Q, 1, f}) \simeq \opCat{\Ho(\CDGA_{\Q, 1, f})} \]
\end{theorem}
}
\Frame{What is a cdga?}{
\begin{definition}
a cdga $A$ is
\begin{itemize}
\item a $\Q$-vector space
\item with a multiplication $A \tensor A \tot{\mu} A$
\item with a differential $A \tot{d} A$ such that $d^2 = 0$
\item with a grading $A = \bigoplus_{n \in \N} A^n$
\item it is commutative: $ x y = (-1)^{\deg{x}\cdot\deg{y}} y x $
\end{itemize}
\end{definition}
}
\Frame{Free cdga's}{
As always: there is a free guy: $\Lambda(...)$
For example
\[ \Lambda(t, dt) \text{ with } \deg{t} = 0 \]
is just polynomials in $t$, with its differential $dt$
}
\newcommand{\Dict}[1]{
\noindent
\begin{tabularx}{\textwidth}{ X X }
{\bf rational spaces} & {\bf cdga's} \\[1em]
#1
\end{tabularx}
}
\Frame{Dictionary}{
\Dict{
$S^n_\Q$ with $n$ odd
& $\Lambda(e)$ with $\deg{e} = n$ \\[1em]
$S^n_\Q$ with $n$ even
& $\Lambda(e, f)$ with $\deg{e} = n$, $\deg{f} = 2n-1$ and $d f = e^2$ \\[1em]
Eilenberg-MacLane space $K(\Q, n)$
& $\Lambda(e)$ with $\deg{e} = n$
}
}
\Frame{Dictionary}{
\Dict{
weak equivalence $$\pi_n(f): \pi_n(X) \iso \pi_n(Y)$$
& weak equivalence $$H(f): H(X) \iso H(Y)$$ \\[1em]
homotopy $$h: X \times I \to Y$$
& homotopy $$h: A \to B \tensor \Lambda(t, dt)$$
}
}
\Frame{Dictionary}{
\Dict{
$$ \pi_n(X) = [S^n, X] $$
& {\begin{align*}
\pi^n(A) &= H(Q(A)) \\
\pi^n(A)^\ast &\iso [A, \Lambda(e)] \\
&\text{or } [A, \Lambda(e, f)]
\end{align*}} \\[1em]
Long exact sequence of a fibration
& Long exact sequence of a cofibration \\[1em]
}
}
\Frame{Dictionary}{
\bf topological $n$-simplex
\[ \Delta^n = \left\{ (x_0, \ldots, x_n) \in \R^{n+1} \,|\, \sum x_i = 1, x_i \geq 0 \right\} \]
\bigskip
\bf cdga $n$-simplex
\[ \Delta_n = \frac{\Lambda(x_0, \ldots x_n, dx_0, \ldots, dx_n)}{\langle \sum x_i - 1, \sum dx_i \rangle}, \quad \deg{x_i}=0 \]
}
\Frame{Construction}{
\begin{center}
\begin{tikzcd}[column sep=huge, row sep=huge, ampersand replacement=\&]
\DELTA \arrow[d, "y"] \arrow[rd, "\Delta_{(-)}"] \& \\
\sSet \arrow[r, dashed, shift left = 1ex, "A"] \& \opCat{\CDGA_\Q} \arrow[l, dashed, shift left = 1ex]
\end{tikzcd}
\end{center}
\bigskip
\pause
\[ A(X) = \Hom_\sSet(X, \Delta_{(-)}) \]
}
\end{document}