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\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}