Adds a section about rationalizations

thesis/notes/Rationalization.tex

@@ -0,0 +1,121 @@
+
+\section{Rationalizations}
+\label{sec:rationalizations}
+
+In this section we will prove the existence of rationalizations $X \to X_\Q$. We will do this in a cellular way. The $n$-spheres play an important role here, so their rationalizations will be discussed first. Again spaces (except for $S^1$) are assumed to be $1$-connected.
+
+\subsection{Construction of \texorpdfstring{$S^n_\Q$}{SnQ}}
+Fix $n>0$ we will construct the rationalization in stages, where at each stage we wedge a sphere and then glue a $n+1$-cell to ``invert'' some element in the $n$th homotopy group. At each stage the space will be homotopy equivalent to $S^n$.
+
+\todo{Put this in a lemma. And make it more readable.}
+We start with $S^n(1) = S^n$. Now assume $S^n(k)$ is constructed. Let $f: S^n \to S^n(k)$ be a representative for $1 \in \Z \iso \pi_n(S^n(k))$ and $g: S^n \to S^n$ be a representative for $k+1 \in \Z \iso \pi_n(S^n)$. These maps combine into $\phi: S^n \to S^n \vee S^n \tot{f \vee g} S^n(k) \vee S^n$. We define $S^n(k+1)$ as the pushout in the following diagram.
+
+\begin{displaymath}
+	\xymatrix{
+	S^n \ar[r]^{\phi} \arcof[d] & S^n(k) \vee S^n \ar[d] \\
+	D^{n+1} \ar[r] & S^n(k+1)
+	}
+\end{displaymath}
+
+Note that $S^n(k+1)$ is homotopy equivalent to $S^n$. More importantly if we identify $\pi_n(S^n(k)) \iso \Z$ and $\pi_n(S^n(k+1)) \iso \Z$, then the inclusions $S^n(k) \subset S^n(k+1)$ induces multiplication by $k+1$ on the homotopy groups.
+
+Now to finish we define the \Def{rational sphere} as $S^n_\Q = \colim_r S^n(r)$. Note that the homotopy groups commute with filtered colimits \cite[9.4]{may}, so that we can compute $\pi_n(S^n_\Q)$ as the colimit of the terms $\pi_n(S^n(r)) \iso \Z$ and the induced maps as depicted in the following diagram:
+
+$$ \Z \tot{\cdots 2} \Z \tot{\cdots 3} \Z \tot{\cdots 4} \Z \tot{\cdots 5} \cdots \Q. $$
+
+Moreover we note that the generator $1 \in \pi_n(S^n)$ is sent to $1 \in \pi_n(S^n_\Q$. However the other homotopy groups are harder to calculate as we have generally no idea how the induced maps will look like. But in the case of $n=1$, the other trivial homotopy groups of $S^1$ are trivial.
+
+\Corollary{rationalization-S1}{
+	The inclusion $S^1 \to S^1_\Q$ is a rationalization.
+}
+
+For $n>1$ we can resort to homology, which also commutes with filtered colimits \cite[14.6]{may}. By connectedness we have $H_0(S^n_\Q) = \Z$ and for $i \neq 0, n$ we have $H_i(S^n) = 0$, so in these cases the homology of the colimit is also $\Z$ and resp. $0$. For $i = n$ we can use the same sequence as above (or use the Hurewicz theorem) to conclude:
+
+$$ H_i(S^n_\Q) = \begin{cases}
+	\Z, &\text{ if } i = 0 \\
+	\Q, &\text{ if } i = n \\
+	0,  &\text{ otherwise.}
+\end{cases} $$
+
+By the Serre-Hurewicz theorem (\TheoremRef{serre-hurewicz}, with $\C$ the class of uniquely divisible groups) we see that $S^n_\Q$ is indeed rational. Then by the Serre-Whitehead theorem (\TheoremRef{serre-whitehead}, with $\C$ the class of torsion groups) the inclusion map $S^n \to S^n_\Q$ is a rationalization.
+
+\Corollary{rationalization-Sn}{
+	The inclusion $S^n \to S^n_\Q$ is a rationalization.
+}
+
+The \Def{rational disk} is now defined as cone of the rational sphere: $D^{n+1}_\Q = CS^n_\Q$. This gives an inclusions of pairs of spaces $(D^{n+1}, S^n) \subset (D^{n+1}_\Q, S^n_\Q)$.
+
+\Lemma{SnQ-extension}{
+	Let $X$ be a rational space and $f : S^n \to X$ be a map. Then this map extends to a map $f' : S^n_\Q \to X$ making the following diagram commute.
+	\begin{displaymath}
+		\xymatrix{
+		S^n \ar[r]^i \ar[rd]^f & S^n_\Q \ar@{-->}[d]^{f'} \\
+		                       & X
+		}
+	\end{displaymath}
+}
+\Proof{
+	Note that $f$ represents a class $\alpha \in \pi_n(X)$. Since $\pi_n(X)$ is a $\Q$-vector space there are elements $\frac{1}{2}\alpha, \frac{1}{3}\alpha, \ldots$ with representatives $\frac{1}{2}f, \frac{1}{3}f, \ldots$. Recall that $S^n_\Q$ consists of many copies of $S^n$, we can define $f'$ on the $k$th copy to be $\frac{1}{k!}f$, as shown in the following diagram.
+
+	\cimage[scale=0.6]{SnQ_Extension}
+
+	Since $[\frac{1}{(k-1)!}f] = k[\frac{1}{k}f] \in \pi_n(X)$ we can extend on the $n+1$-cells. This defines $f'$. Since our inclusion $i: S^n \cof S^n_\Q$ is in the first sphere, we get $f = f' \circ i$.
+}
+\todo{Add: unique up to homotopy. A homotopy extends to a homotopy.}
+
+\subsection{Rationalizations of arbitrary spaces}
+Having rational cells we wish to replace the cells in a CW complex $X$ by the rational cells to obtain a rationalization.
+
+\Lemma{rationalization-CW}{
+	Any CW complex admits a rationalization.
+}
+\Proof{
+	Let $X$ be a CW complex. We will define $X_\Q$ with induction on the dimension of the cells. Since $X$ is simply connected we can start with $X^0_\Q = X^1_\Q = \ast$. Now assume that the rationalization $X^k \tot{\phi^k} X^k_\Q$ is already defined. Let $A$ be the set of $k+1$-cells and $f_\alpha : S^k \to X^{k+1}$ be the attaching maps. Then by \LemmaRef{SnQ-extension} these extend to $g_\alpha = (\phi^k \circ f_\alpha)' : S^k_\Q \to X^k_\Q$. This defines $X^{k+1}_\Q$ as the pushout in the following diagram.
+
+	\begin{displaymath}
+		\xymatrix{
+		\coprod_A S^n_\Q \ar[r]^{(g_\alpha)} \arcof[d] \xypo  &  X^k_\Q \ar@{-->}[d] \\
+		\coprod_A D^{n+1}_\Q \ar@{-->}[r] & X^{k+1}_\Q
+		}
+	\end{displaymath}
+
+	Now by the universal property of $X^{k+1}$, we get a map $\phi^{k+1} : X^{k+1} \to X^{k+1}_\Q$ which is compatible with $\phi^k$ and which is a rationalization.
+}
+
+\todo{For arbitrary spaces}
+
+\Theorem{}{
+	The above construction is in fact a \Def{localization}, i.e. for any map $f : X \to Z$ to a rational space $Z$, there is an extension $f' : X_\Q \to Z$ making the following diagram commute.
+
+	\begin{displaymath}
+		\xymatrix{
+		X \ar[r]^i \ar[rd]^f & X_\Q \ar@{-->}[d]^{f'} \\
+		                       & Z
+		}
+	\end{displaymath}
+
+	Moreover, any $f''$ making the diagram commute is homotopic to $f'$ and if $g : X \to Z$ is homotopic to $f$ then the extension $g'$ is homotopic to $f'$.
+}
+
+The extension property allows us to define a rationalization of maps. Given $f : X \to Y$, we can consider the composite $if : X \to Y \to Y_\Q$. Now this extends to $(if)' : X_\Q \to Y_\Q$. Note that this construction is not functorial, since there are choices of homotopies involved. When passing to the homotopy category, however, this construction \emph{is} functorial and has an universal property.
+
+We already mentioned in the first section that for rational spaces the notions of weak equivalence and rational equivalence coincide. Now that we always have a rationalization we have:
+
+\Corollary{}{
+	Let $f: X \to Y$ be a map, then $f$ is a rational equivalence if and only if $f_\Q : X_\Q \to Y_\Q$ is a weak equivalence.
+}
+
+\Corollary{}{
+	The homotopy category of $1$-connected rational spaces is equivalent to the rational homotopy category of $1$-connected spaces.
+}
+
+\subsection{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.
+
+\Lemma{rationalization-em-space}{
+	Let $A$ be an abelian group and $n \geq 1$. Then
+	$$ K(A, n) \to K(A \tensor \Q, n) $$
+	is a rationalization
+}
+
+Postnikov

thesis/notes/Serre.tex

@@ -86,7 +86,7 @@ For the main theorem we need the following construction. \todo{Geef de construct
 	\end{itemize}
 }
 
-\Theorem{absolute-serre-hurewicz}{
+\Theorem{serre-hurewicz}{
 	(Absolute Serre-Hurewicz Theorem)
 	Let $\C$ be a Serre class. Let $X$ a $1$-connected space.
 	If $\pi_i(X) \in C$ for all $i<n$, then $H_i(X) \in C$ for all $i<n$ and the Hurewicz map $h: \pi_i(X) \to H_i(X)$ is a $\C$-isomorphism for all $i \leq n$.

thesis/thesis.tex

@@ -41,6 +41,7 @@ Some categories:
 \addtocontents{toc}{\protect\setcounter{tocdepth}{2}}
 \myinput{notes/Basics}
 \myinput{notes/Serre}
+\myinput{notes/Rationalization}
 \myinput{chapters/Homotopy_Theory_CDGA}
 \myinput{chapters/Polynomial_Forms}
 \myinput{notes/Minimal_Models}