Fixes some small things and adds some todos

thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex

@@ -8,9 +8,9 @@ Recall that a cdga $A$ is a commutative differential graded algebra, meaning tha
 	\item it has an associative and unital multiplication: $\mu: A \tensor A \to A$ and
 	\item it is commutative: $x y = (-1)^{\deg{x}\cdot\deg{y}} y x$.
 \end{itemize}
-And all of the above structure is compatible with each other (e.g. the differential is a derivative, the maps are graded, \dots). We have a left adjoint $\Lambda$ to the forgetful functor $U$ which assigns the free graded commutative algebras $\Lambda V$ to a graded module $V$. This extends to an adjunction (also called $\Lambda$ and $U$) between commutative differential graded algebras and differential graded modules.
+And all of the above structure is compatible with each other (e.g. the differential is a derivation of degree $1$, the maps are graded, \dots). We have a left adjoint $\Lambda$ to the forgetful functor $U$ which assigns the free graded commutative algebras $\Lambda V$ to a graded module $V$. This extends to an adjunction (also called $\Lambda$ and $U$) between commutative differential graded algebras and differential graded modules.
 
-In homological algebra we are especially interested in \emph{quasi isomorphisms}, i.e. the maps $f: A \to B$ inducing an isomorphism on homology: $H(f): HA \iso HB$. This notions makes sense for any object with a differential.
+In homological algebra we are especially interested in \emph{quasi isomorphisms}, i.e. the maps $f: A \to B$ inducing an isomorphism on cohomology: $H(f): HA \iso HB$. This notions makes sense for any object with a differential.
 
 We furthermore have the following categorical properties of cdga's:
 \begin{itemize}

thesis/notes/Basics.tex

@@ -4,7 +4,7 @@
 
 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 call a space \Def{simple} if it is connected and its fundamental group is abelian.
+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.\todo{non-standard}
 
 \Definition{rational-space}{
 	A simple space $X$ is a \emph{rational space} if
@@ -30,7 +30,7 @@ Note that for a rational space $X$, the ordinary homotopy groups are isomorphic
 
 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.
 
-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).
+The theory of rational homotopy 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}

thesis/notes/CDGA_Of_Polynomials.tex

@@ -1,10 +1,10 @@
 
-We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows for simplicial methods. In the following definition one should be reminded of the topological $n$-simplex defined as convex span.
+We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows for simplicial methods. In the following definition one should remember the topological $n$-simplex defined as convex span.
 
 \Definition{apl}{
 	For all $n \in \N$ define the following cdga:
-	$$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, d x_0, \ldots, d x_n)}{(\sum_{i=0}^n x_i - 1, \sum_{i=0}^n d x_i)} $$
+	$$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, d x_0, \ldots, d x_n)}{(\sum_{i=0}^n x_i - 1, \sum_{i=0}^n d x_i)}, $$
-	So it is the free cdga with $n+1$ generators and their differentials such that $\sum_{i=0}^n x_i = 1$ and in order to be well behaved $\sum_{i=0}^n d x_i = 0$.
+	where $\deg{x_i} = 0$. So it is the free cdga with $n+1$ generators and their differentials such that $\sum_{i=0}^n x_i = 1$ and in order to be well behaved $\sum_{i=0}^n d x_i = 0$.
 }
 
 Note that the inclusion $\Lambda(x_1, \ldots, x_n, d x_1, \ldots, d x_n) \to \Apl_n$ is an isomorphism of cdga's. So $\Apl_n$ is free and (algebra) maps from it are determined by their images on $x_i$ for $i = 1, \ldots, n$ (also note that this determines the images for $d x_i$). This fact will be used throughout.

thesis/notes/Homotopy_Relations_CDGA.tex

@@ -1,7 +1,7 @@
 
 Although the abstract theory of model categories gives us tools to construct a homotopy relation (\DefinitionRef{homotopy}), it is useful to have a concrete notion of homotopic maps.
 
-Consider the free cdga on one generator $\Lambda(t, dt)$, where $\deg{t} = 0$, this can be thought of as the (dual) unit interval with endpoints $1$ and $t$. We define two \emph{endpoint maps} as follows:
+Consider the free cdga on one generator $\Lambda(t, dt)$\todo{same as $\Lambda D(0)$}, where $\deg{t} = 0$, this can be thought of as the (dual) unit interval with endpoints $1$ and $t$. We define two \emph{endpoint maps} as follows:
 $$ d_0, d_1 : \Lambda(t, dt) \to \k $$
 $$ d_0(t) = 1, \qquad d_1(t) = 0, $$
 this extends linearly and multiplicatively. Note that it follows that we have $d_0(1-t) = 0$ and $d_1(1-t) = 1$. These two functions extend to tensor products as $d_0, d_1: \Lambda(t, dt) \tensor X \to \k \tensor X \tot{\iso} X$.
@@ -12,7 +12,7 @@ this extends linearly and multiplicatively. Note that it follows that we have $d
 	such that $d_0 h = g$ and $d_1 h = f$.
 }
 
-In terms of model categories, such a homotopy is a right homotopy and the object $\Lambda(t, dt) \tensor X$ is a path object for $X$. We can easily see that it is a very good path object. First note that $\Lambda(t, dt) \tensor X \tot{(d_0, d_1)} X \oplus X$ is surjective (for $(x, y) \in X \oplus X$ take $t \tensor x + (1-t) \tensor y$). Secondly we note that $\Lambda(t, dt) = \Lambda(D(0))$ and hence $\k \to \Lambda(t, dt)$ is a cofibration, by \LemmaRef{model-cats-coproducts} we have that $X \to \Lambda(t, dt) \tensor X$ is a (necessarily trivial) cofibration.
+In terms of model categories, such a homotopy is a right homotopy and the object $\Lambda(t, dt) \tensor X$ is a path object for $X$. We can easily see that it is a very good path object\todo{Refereer}. First note that $\Lambda(t, dt) \tensor X \tot{(d_0, d_1)} X \oplus X$ is surjective (for $(x, y) \in X \oplus X$ take $t \tensor x + (1-t) \tensor y$). Secondly we note that $\Lambda(t, dt) = \Lambda(D(0))$ and hence $\k \to \Lambda(t, dt)$ is a cofibration, by \LemmaRef{model-cats-coproducts} we have that $X \to \Lambda(t, dt) \tensor X$ is a (necessarily trivial) cofibration.
 
 Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{right_homotopy}), however the converse need not be true.
 

thesis/notes/Model_Of_CDGA.tex

@@ -93,7 +93,7 @@ We can use Quillen's small object argument with these sets. The argument directl
 	[MC5a] A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is a trivial cofibration and $p$ a fibration.
 \end{corollary}
 
-The previous factorization can also be described explicitly as seen in \cite{bousfield}. Let $f: A \to X$ be a map, define $E = A \tensor \bigtensor_{x \in X}T(\deg{x})$. Then $f$ factors as:
+The previous factorization can also be described explicitly as seen in \cite{bousfield}. Let $f: A \to X$ be a map, define $E = A \tensor \bigtensor_{x \in X}T(\deg{x})$. Then $f$ factors as:\todo{This is later defined as $A \tensor \Lambda(C(X))$, which is precisely the same}
 $$ A \tot{i} E \tot{p} X, $$
 where $i$ is the obvious inclusion $i(a) = a \tensor 1$ and $p$ maps (products of) generators $a \tensor b_x$ with $b_x \in T(\deg{x})$ to $f(a) \cdot x \in X$.
 

thesis/notes/Polynomial_Forms.tex

@@ -1,5 +1,5 @@
 
-There is a general way to construct functors from $\sSet$ whenever we have some simplicial object. In our case we have the simplicial cdga $\Apl$ (which is nothing more than a functor $\opCat{\DELTA} \to \CDGA$) and we want to extend to a contravariant functor $\sSet \to \CDGA_\k$. This will be done via \Def{Kan extensions}.
+There is a general way to construct contravariant functors from $\sSet$ whenever we have some simplicial object. In our case we have the simplicial cdga $\Apl$ (which is nothing more than a functor $\opCat{\DELTA} \to \CDGA$) and we want to extend to a contravariant functor $\sSet \to \CDGA_\k$. This will be done via \Def{Kan extensions}.
 
 Given a category $\cat{C}$ and a functor $F: \DELTA \to \cat{C}$, then define the following on objects:
 \begin{align*}

thesis/notes/Rationalization.tex

@@ -5,9 +5,9 @@
 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. In this section $1$-connectedness of spaces will play an important role.
 
 \section{Rationalization of \texorpdfstring{$S^n$}{Sn}}
-In this section we fix $n>0$. We will construct $S^n_\Q$ in stages $S^n(1), S^n(2), \ldots$, 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.
+In this section we fix $n>0$. We will construct $S^n_\Q$ in stages $S^n(1), S^n(2), \ldots$, where at each stage we wedge a sphere and then glue an $n+1$-cell to ``invert'' some element in the $n$th homotopy group.
 
-\todo{plaatje}
+\todo{plaatje en tekst leesbaarder maken}
 
 We start the construction with $S^n(1) = S^n$. Assume we constructed $S^n(r) = \bigvee_{i=1}^{r} S^{n} \cup_{h} \coprod_{i=1}^{r-1} D^{n+1}$, where $h$ is a specific attaching map. Assume furthermore the following two properties. Firstly, the inclusion $i_r : S^n \to S^n(r)$ of the terminal sphere is a weak equivalence. Secondly, the inclusion $i_1 : S^n \to S^n(r)$ of the initial sphere induces the multiplication $\pi_n(S^n) \tot{\times r!} \pi_n(S^n(r))$ under the identification of $\pi_n(S^n) = \pi_n(S^n(r)) = \Z$.
 
@@ -25,7 +25,7 @@ Now to finish the construction we define the \Def{rational sphere} as $S^n_\Q =
 
 $$ \Z \tot{\times 2} \Z \tot{\times 3} \Z \tot{\times 4} \Z \tot{\times 5} \cdots \Q. $$
 
-Moreover we note that the generator $1 \in \pi_n(S^n)$ is sent to $1 \in \pi_n(S^n_\Q)$ via the inclusion $S^n \to S^n_\Q$ of the initial sphere. 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.
+Moreover we note that the generator $1 \in \pi_n(S^n)$ is sent to $1 \in \pi_n(S^n_\Q)$ via the inclusion $S^n \to S^n_\Q$ of the initial sphere. However the other homotopy groups are harder to calculate as we have generally no idea what the induced maps are. But in the case of $n=1$, the other homotopy groups of $S^1$ are trivial.
 
 \Corollary{rationalization-S1}{
 	The inclusion $S^1 \to S^1_\Q$ is a rationalization.
@@ -81,7 +81,7 @@ Having rational cells we wish to replace the cells in a CW complex $X$ by the ra
 	Any simply connected 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 pullback in the following diagram.
+	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{
@@ -137,7 +137,7 @@ There are others ways to obtain a rationalization. One of them relies on the obs
 	is a rationalization
 }
 
-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
+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 pullback in
 \begin{displaymath}
 	\xymatrix{
 	P_{n}(X) \ar[r] \arfib[d] \xypb & PK(\pi_n(X), n+1) \arfib[d] \\

thesis/notes/Serre.tex

@@ -9,7 +9,7 @@ In this section we will prove the Whitehead and Hurewicz theorems in a rational
 		\item for all exact sequences $0 \to A \to B \to C \to 0$ if two abelian groups are in $\C$, then so is the third,
 		\item for all $A \in \C$ the tensor product $A \tensor B$ is in $\C$ for any abelian group $B$,
 		\item for all $A \in \C$ the Tor group $\Tor(A, B)$ is in $\C$ for any abelian group $B$, and
-		\item for all $A \in \C$ the group-homology $H_i(A; \Z)$ is in $\C$ for all positive $i$.
+		\item for all $A \in \C$ the group homology $H_i(A; \Z)$ is in $\C$ for all positive $i$.
 	\end{itemize}
 }