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}
In the following definition \emph{space} is to be understood as a topological space or a simplicial set. We will call a space \Def{simple} if it is connected and its fundamental group is abelian.
\Definition{rational-space}{
A space $X$ is a \emph{rational space} if
A simple 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:
We define the \emph{rational homotopy groups} of a simple 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. $\pi_i(X)\tensor\Q\iso\pi_i(X)$.
In order to define the tensor product $\pi_1(X)\tensor\Q$ we need that the fundamental group is abelian, that is the rational homotopy groups are only defined for simple spaces. There is a more general approach using \Def{nilpotent groups}, which admit $\Q$-completions \cite{bousfield}. Since this is rather technical we will often restrict ourselves to simple spaces or even simply connected spaces.
Note that for a rational space $X$, the ordinary 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 $i > 0$.
@ -28,16 +30,17 @@ Note that for a rational space $X$, the homotopy groups are isomorphic to the ra
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}
The theory of rational homotopy theory is the study of simple spaces with rational equivalences. Quillen defines a model structure on simply connected simplicial sets with rational equivalences as weak equivalences \cite{Quillen}. This means that there is a homotopy category $\Ho^\Q(\sSet_1)$. However we will later prove that every simply connected space has a rationalization, so that $\Ho^\Q(\sSet_1)=\Ho(\sSet^\Q_1)$ are equivalent categories. This means that we do not need the model structure defined by Quillen, but we can simply restrict ourselves to rational spaces (with ordinary weak equivalences).
\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.
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}.
\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 $i \leq n +1$.
(Relative Hurewicz) For any inclusion of spaces $Y\subset X$ and all $i > 0$, there is a natural map
$$ h_i : \pi_i(X, Y)\to H_i(X, Y). $$
If furthermore $(X, Y)$ is $n$-connected ($n > 0$), then the map $h_i$ is an isomorphism for all $i \leq n +1$.
}
\Theorem{serre-les}{
@ -82,9 +85,8 @@ The latter two theorems have a direct consequence for rational homotopy theory.
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:
Let $f: X \to Y$ be a Serre fibration of simple spaces with a simple fiber, then there is a natural long exact sequence of rational homotopy groups:
@ -130,12 +130,30 @@ We already mentioned in the first section that for rational spaces the notions o
}
\section{Other constructions}
There are others ways to obtain a rationalization. One of them relies on the observations that it is easy to rationalize Eilenberg-MacLane spaces.
There are others ways to obtain a rationalization. One of them relies on the observations that it is easy to rationalize Eilenberg-MacLane spaces. Since we already have a rationalization at hand the details in this section will be skipped and the focus lies on the construction.
\Lemma{rationalization-em-space}{
\Remark{rationalization-em-space}{
Let $A$ be an abelian group and $n \geq1$. Then
$$ K(A, n)\to K(A \tensor\Q, n)$$
is a rationalization
}
Postnikov
Any simply connected space can be decomposed into a Postnikov tower $X \to\ldots\fib P_2(X)\fib P_1(X)\fib P_0(X)$\cite[Chapter 22.4]{may}. Furthermore if $X$ is a simply connected CW complex, $P_{n}(X)$ can be constructed from $P_{n-1}(X)$ as the pushout in
where the map $k_{n-1}$ is called the $k$-invariant. We will only need its existence for the construction. The rationalization can now be constructed with induction on this Postnikov tower. Start the induction with $X_\Q(2)= K(\pi_2(X)\tensor\Q, 2)$. Now assume we constructed $X_\Q(r-1)$ compatible with the $k$-invariant described above. We are in the following situation:
where the bottom square is our induction hypothesis, the right square is by naturality of the path space fibration and the back face is the pullback described above. We can define $X_\Q(r)$ to be the pullback of the front face, which induces a map $\phi_r : P_r(X)\to X_\Q(r)$. By inspecting the long exact sequence of the fibration $X_\Q(r)\fib X_\Q(r-1)$ we see that $\phi_r$ is indeed a rationalization.
We finish the construction by defining $X_\Q=\lim_r X_\Q(r)$. For more details, one can read \cite{sullivan} or \cite{berglund}.
Joshua Moerman
10 years ago