diff --git a/thesis/images/SnQ_Extension.png b/thesis/images/SnQ_Extension.png new file mode 100644 index 0000000..e8d1ba9 Binary files /dev/null and b/thesis/images/SnQ_Extension.png differ diff --git a/thesis/notes/Rationalization.tex b/thesis/notes/Rationalization.tex new file mode 100644 index 0000000..238f354 --- /dev/null +++ b/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 diff --git a/thesis/notes/Serre.tex b/thesis/notes/Serre.tex index 96286f7..c938494 100644 --- a/thesis/notes/Serre.tex +++ b/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