\chapter{Rational homotopy theory}
\label{sec:basics}

In this section we will state the aim of rational homotopy theory. Moreover we will recall classical theorems from algebraic topology and deduce rational versions of them.

In the following definition \emph{space} is to be understood as a topological space or a simplicial set. We will restrict ourselves to simply connected spaces. \todo{Per definitie/stelling samenhangendheid aangeven}

\Definition{rational-space}{
	A space $X$ is a \emph{rational space} if
	$$ \pi_i(X) \text{ is a $\Q$-vector space } \quad\forall i > 0. $$
}

\Definition{rational-homotopy-groups}{
	We define the \emph{rational homotopy groups} of a space $X$ as:
	$$ \pi_i(X) \tensor \Q \quad \forall i > 0.$$
}

Note that for a rational space $X$, the homotopy groups are isomorphic to the rational homotopy groups, i.e. $\pi_i(X) \tensor \Q \iso \pi_i(X)$.

\Definition{rational-homotopy-equivalence}{
	A map $f: X \to Y$ is a \emph{rational homotopy equivalence} if $\pi_i(f) \tensor \Q$ is a linear isomorphism for all $i > 0$.
}

\Definition{rationalization}{
	A map $f: X \to X_0$ is a \emph{rationalization} if $X_0$ is rational and $f$ is a rational homotopy equivalence.
}

Note that a weak equivalence (and hence also a homotopy equivalence) is always a rational homotopy theory. Furthermore if $f: X \to Y$ is a map between rational spaces, then $f$ is a rational homotopy equivalence if and only if $f$ is a weak equivalence.

We will later see that any space admits a rationalization. The theory of rational homotopy theory is then the study of the homotopy category $\Ho_\Q(\Top) \iso \Ho(\Top_\Q)$, which is on its own turn equivalent to $\Ho(\sSet_\Q) \iso \Ho_\Q(\sSet)$. \todo{Notatie}

\section{Classical results from algebraic topology}

We will now recall known results from algebraic topology, without proof. One can find many of these results in basic text books, such as \cite{may, dold}. We do not assume $1$-connectedness here.

\Theorem{relative-hurewicz}{
	(Relative Hurewicz) For any inclusion of spaces $A \subset X$ and all $i > 0$, there is a natural map
	$$ h_i : \pi_i(X, A) \to H_i(X, A). $$ \todo{Andere letter dan $A$}
	If furthermore $(X,A)$ is $n$-connected ($n > 0$), then the map $h_i$ is an isomorphism for all $i \leq n + 1$.
}

\Theorem{serre-les}{
	(Long exact sequence) Let $f: X \to Y$ be a Serre fibration, then there is a long exact sequence (note that $X$ and $Y$ need not be $1$-connected):
	$$ \cdots \tot{\del} \pi_i(F) \tot{i_\ast} \pi_i(X) \tot{f_\ast} \pi_i(Y) \tot{\del} \cdots \to \pi_0(Y) \to \ast, $$
	where $F$ is the fiber of $f$.
}

Using an inductive argument and the previous two theorems, one can show the following theorem (as for example shown in \cite{griffiths}).
\Theorem{whitehead-homology}{
	(Whitehead) For any map $f: X \to Y$ between $1$-connected spaces, $ \pi_i(f) $ is an isomorphism $\forall 0 < i < r$ if and only if $H_i(f)$ is an isomorphism $\forall 0 < i < r$.
	In particular we see that $f$ is a weak equivalence if and only if it induces an isomorphism on homology.
}

The following two theorems can be found in textbooks about homological algebra such as \cite{weibel, rotman}. Note that when the degrees are left out, $H(X; A)$ denotes the graded homology module with coefficients in $A$.

\Theorem{universal-coefficient}{
	(Universal Coefficient Theorem)
	For any space $X$ and abelian group $A$, there are natural short exact sequences
	$$ 0 \to H_n(X) \tensor A \to H_n(X; A) \to \Tor(H_{n-1}(X), A) \to 0, $$
	$$ 0 \to \Ext(H_{n-1}(X), A) \to H^n(X; A) \to \Hom(H_n(X), A) \to 0. $$
}

\Theorem{kunneth}{
	(Künneth Theorem)
	For spaces $X$ and $Y$, there is a short exact sequence
	$$ 0 \to H(X; A) \tensor H(Y; A) \to H(X \times Y; A) \to \Tor(H(X; A), H(Y; A)) \to 0, $$
	where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded. \todo{Geef algebraische versie voor ketencomplexen}
}

\section{Immediate results for rational homotopy theory}

The latter two theorems have a direct consequence for rational homotopy theory. By taking $A = \Q$ we see that the torsion groups vanish. We have the immediate corollary.

\Corollary{rational-corollaries}{
	We have the following natural isomorphisms in homology
	$$ H(X) \tensor \Q \tot{\iso} H(X; \Q), $$
	$$ H(X \times Y; \Q) \tot{\iso} H(X; \Q) \tensor H(Y; \Q). $$
	Furthermore we can relate homology and cohomology in a natural way:
	$$ H^n(X; \Q) \tot{\iso} \Hom(H_n(X); \Q). $$
}

The long exact sequence for a Serre fibration also has a direct consequence for rational homotopy theory.
\Corollary{rational-les}{
	Let $f: X \to Y$ be a Serre fibration of $1$-connected spaces, then there is a natural long exact sequence of rational homotopy groups:
	$$ \cdots \tot{\del} \pi_i(F) \tensor \Q \tot{i_\ast} \pi_i(X) \tensor \Q \tot{f_\ast} \pi_i(Y) \tensor \Q \tot{\del} \cdots. $$
	\todo{Wat als $F$ niet $1$-connected is? En: $\pi_1$ abels is ook goed.}
}

In the next sections we will prove the rational Hurewicz and rational Whitehead theorems. These theorems are due to Serre \cite{serre}.