Fixes small (style) errors

thesis/chapters/Introduction.tex

@@ -5,24 +5,16 @@ Schrijf hier wat
 
 \section{Preliminaries and Notation}
 
-We assume the reader is familiar with category theory, basics from algebraic topology and the basics of simplicial sets. Some knowledge about differential graded algebra (or homological algebra) and model categories is assumed, but the reader may review this in the appendices.
+We assume the reader is familiar with category theory, basics from algebraic topology and the basics of simplicial sets. Some knowledge about differential graded algebra (or homological algebra) and model categories is assumed, but the reader may review some facts on this in the appendices.
 
+We will fix the following notations and categories.
 \begin{itemize}
-	\item $\k$ will denote an arbitrary commutative ring (or field, if indicated at the start of a section). Modules, tensor products, \dots are understood as $\k$-modules, tensor products over $\k$, \dots. If ambiguity can occur notation will be explicit.
+	\item $\k$ will denote an arbitrary commutative ring (or field, if indicated at the start of a section). Modules, tensor products, \dots are understood as $\k$-modules, tensor products over $\k$, \dots.
-	\item $\cat{C}$ will denote an arbitrary category.
+	\item $\Hom_{\cat{C}}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ may occasionally be left out.
-	\item $\cat{0}$ (resp. $\cat{1}$) will denote the initial (resp. final) objects in a category $\cat{C}$.
+	\item $\Top$: category of topological spaces and continuous maps. We denote the full subcategory of $r$-connected spaces by $\Top_r$, this convention is also used for other categories.
-	\item $\Hom_{\cat{C}}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ is occasionally left out if the category is clear from the context.
-\end{itemize}
-
-Some categories:
-\begin{itemize}
-	\item $\Top$: category of topological spaces and continuous maps.
 	\item $\Ab$: category of abelian groups and group homomorphisms.
-	\item $\DELTA$: category of simplices (i.e. finite, non-empty ordinals) and order preserving maps.
+	\item $\sSet$: category of simplicial sets and simplicial maps (more generally we have the category of simplicial objects, $\cat{sC}$, for any category $\cat{C}$). We have the homotopy equivalence $|-| : \sSet \leftadj \Top : S$.
-	\item $\sSet$: category of simplicial sets and simplicial maps (more generally we have the category of simplicial objects, $\cat{sC}$, for any category $\cat{C}$).
+	\item $\DGA_\k$: category of non-negatively differential graded algebras over $\k$ (as defined in the appendix) and graded algebra  maps. As a shorthand we will refer to such an object as \emph{dga}. Furthermore $\CDGA_\k$ is the full subcategory of $\DGA_\k$ of commutative dga's (\emph{cdga}'s).
-	\item $\Ch{\k}, \CoCh{\k}$: category of non-negatively graded chain (resp. cochain) complexes and chain maps.
-	\item $\DGA_\k$: category of non-negatively differential graded algebras over $\k$ (these are cochain complexes with a multiplication) and graded algebra  maps. As a shorthand we will refer to such an object as \emph{dga}.
-	\item $\CDGA_\k$: the full subcategory of $\DGA_\k$ of commutative dga's (\emph{cdga}'s).
 \end{itemize}
 
 \tableofcontents

thesis/diagrams/Kreck_Exact_Sequence.tex

@@ -1,5 +1,5 @@
 \small
 \xymatrix{
-	\cdots \ar[r] & H_{i+1}(E^{k+1}, E^k) \ar[r] \ar[d]^\iso & H_i(E^k, F) \ar[r] \ar[d] & H_i(E^{k+1}, F) \ar[r] \ar[d] & \cdots \\
+	\cdots \ar[r] & H_{i+1}(E^{k+1}, E^k) \ar[r] \ar[d] & H_i(E^k, F) \ar[r] \ar[d] & H_i(E^{k+1}, F) \ar[r] \ar[d] & \cdots \\
 	\cdots \ar[r] & H_{i+1}(B^{k+1}, B^k) \ar[r] & H_i(B^k, b_0) \ar[r] & H_i(B^{k+1}, b_0) \ar[r] & \cdots
 }

thesis/notes/Basics.tex

@@ -4,19 +4,19 @@
 
 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.\todo{non-standard}
+In the following definition \emph{space} is to be understood as a topological space or a simplicial set.
 
 \Definition{rational-space}{
-	A simple space $X$ is a \emph{rational space} if
+	A $0$-connected space $X$ with abelian fundamental group is a \emph{rational space} if
-	$$ \pi_i(X) \text{ is a $\Q$-vector space } \quad\forall i > 0. $$
+	$$ \pi_i(X) \text{ is a $\Q$-vector space } \quad \forall i > 0. $$
 }
 
 \Definition{rational-homotopy-groups}{
-	We define the \emph{rational homotopy groups} of a simple space $X$ as:
+	We define the \emph{rational homotopy groups} of a $0$-connected space $X$ with abelian fundamental group as:
 	$$ \pi_i(X) \tensor \Q \quad \forall i > 0.$$
 }
 
-In order to define the tensor product $\pi_1(X) \tensor \Q$ we need that the fundamental group is abelian, that is why the definition requires 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.
+In order to define the tensor product $\pi_1(X) \tensor \Q$ we need that the fundamental group is abelian, the higher homotopy groups are always abelian. 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 such 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. $\pi_i(X) \tensor \Q \iso \pi_i(X)$.
 
@@ -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 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 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}
@@ -44,7 +44,7 @@ We will now recall known results from algebraic topology, without proof. One can
 }
 
 \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):
+	(Long exact sequence) Let $f: X \to Y$ be a Serre fibration, then there is a long exact sequence:
 	$$ \cdots \tot{\del} \pi_i(F) \tot{i_\ast} \pi_i(X) \tot{f_\ast} \pi_i(Y) \tot{\del} \cdots \to \pi_0(Y) \to \ast, $$
 	where $F$ is the fiber of $f$.
 }
@@ -89,7 +89,7 @@ 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 simple spaces with a simple fiber, then there is a natural long exact sequence of rational homotopy groups:
+	Let $f: X \to Y$ be a Serre fibration with fiber $F$, all $0$-connected with abelian fundamental group, then there is a natural long exact sequence of rational homotopy groups:
 	$$ \cdots \tot{\del} \pi_i(F) \tensor \Q \tot{i_\ast} \pi_i(X) \tensor \Q \tot{f_\ast} \pi_i(Y) \tensor \Q \tot{\del} \cdots. $$
 }
 

thesis/notes/Model_Of_CDGA.tex

@@ -25,7 +25,7 @@ Note that with these classes, every cdga is a fibrant object.
 	[MC2] The \emph{2-out-of-3} property for quasi isomorphisms.
 \end{lemma}
 \begin{proof}
-	Let $f$ and $g$ be two maps such that two out of $f$, $g$ and $fg$ are weak equivalences. This means that two out of $H(f)$, $H(g)$ and $H(f)H(g)$ are isomorphisms. The \emph{2-out-of-3} property holds for isomorphisms, proving the statement.
+	Let $f$ and $g$ be two maps such that two out of $f$, $g$ and $fg$ are weak equivalences. This means that two out of $H(f)$, $H(g)$ and $H(f)H(g)$ are isomorphisms. The 2-out-of-3 property holds for isomorphisms, proving the statement.
 \end{proof}
 
 \begin{lemma}

thesis/notes/Serre.tex

@@ -13,7 +13,7 @@ In this section we will prove the Whitehead and Hurewicz theorems in a rational
 	\end{itemize}
 }
 
-Serre gave weaker axioms for his classes and proves some of the following lemmas only using these weaker axioms. However the classes we are interested in do satisfy the above (stronger) requirements. One should think of a Serre class as a class of groups we want to \emph{ignore}. We will be interested in the first two of the following examples.
+Serre gave weaker axioms for his classes and proves some of the following lemmas only using these weaker axioms. However the classes we are interested in do satisfy the above (stronger) requirements. One should think of a Serre class as a class of groups we want to \emph{ignore}.
 
 \Example{serre-classes}{
 	We give three Serre classes without proof.
@@ -30,7 +30,7 @@ As noted by Hilton in \cite{hilton} we think of Serre classes as a generalized 0
 	Let $\C$ be a Serre class and let $f: A \to B$ be a map of abelian groups. Then $f$ is a \Def{$\C$-isomorphism} if both the kernel and cokernel lie in $\C$.
 }
 
-Note that the maps $0 \to C$ and $C \to 0$ are $\C$-isomorphisms for any $C \in \C$. More importantly the 5-lemma also holds for $\C$-isos and whenever $f$, $g$ and $g \circ f$ are maps such that two of them are $\C$-iso, then so is the third.
+Note that the maps $0 \to C$ and $C \to 0$ are $\C$-isomorphisms for any $C \in \C$. More importantly the 5-lemma also holds for $\C$-isos and we have the 2-out-of-3 property: whenever $f$, $g$ and $g \circ f$ are maps such that two of them are $\C$-iso, then so is the third.
 
 In the following arguments we will consider fibrations and need to compute homology thereof. Unfortunately there is no long exact sequence for homology of a fibration, however the following lemma expresses something similar. It is usually proven with spectral sequences,  \cite[Ch. 2 Thm 1]{serre}. However in \cite{kreck} we find a more geometric proof.
 
@@ -42,7 +42,7 @@ In the following arguments we will consider fibrations and need to compute homol
 	\end{itemize}
 }
 \Proof{
-	We will first replace the fibration by a fiber bundle. This is done by going to simplicial sets and replace the induced map by a minimal fibration \cite{joyal}. The fibration $p$ induces a fibration $S(E) \tot{S(p)} S(B)$, which can be factored as $S(E) \we M \fib S(B)$, where the map $M \fib S(B)$ is minimal (and hence a fiber bundle). By realizing we obtain the following diagram:
+	We will first replace the fibration by a fiber bundle. This is done by going to simplicial sets and replace the induced map by a minimal fibration as follows. The fibration $p$ induces a fibration $S(E) \tot{S(p)} S(B)$, which can be factored as $S(E) \we M \fib S(B)$, where the map $M \fib S(B)$ is minimal and hence a fiber bundle \cite{joyal}. By realizing we obtain the following diagram:
 	\begin{displaymath}
 		\xymatrix {
 		{|M|} \arfib[d] & \arwe[l] {|S(E)|} \arwe[r] \arfib[d] & E \arfib[d] \\
@@ -54,7 +54,6 @@ In the following arguments we will consider fibrations and need to compute homol
 	So we can assume $E$ and $B$ to be a CW complexes and $E \fib B$ to be a fiber bundle. We will do induction on the skeleton $B^k$. By connectedness we can assume $B^0 = \{ b_0 \}$. Restrict $E$ to $B^k$ and note $E^0 = F$. Now the base case is clear: $H_i(E^0, F) \to H_i(B^0, b_0)$ is a $\C$-iso.
 
 	For the induction step, consider the long exact sequence in homology for the triples $(E^{k+1}, E^k, F)$ and $(B^{k+1}, B^k, b_0)$:
-
 	\cdiagram{Kreck_Exact_Sequence}
 
 	The morphism in the middle is a $\C$-iso by induction. We will prove that the left morphism is a $\C$-iso which implies by the five lemma that the right morphism is one as well.
@@ -72,7 +71,7 @@ In the following arguments we will consider fibrations and need to compute homol
 
 	For $p''$ a similar reasoning holds, it is clearly surjective and we only need to prove that the kernel of $p''$ (which is the Tor group itself) is in $\C$. First notice that $\Tor(H_i(B^{k+1}, B^k), H_0(F)) = 0$ as $H_0(F) \iso \Z$. Then consider the other terms of the graded Tor group. Again we use the assumed bounds to conclude that the Tor group is in $\C$ for $i \leq n+m$. So indeed $p''$ is a $\C$-iso for all $i \leq n+m$.
 
-	Now we conclude that $p_\ast : H_{i+1}(B^{k+1}, B^k) \to H_{i+1}(E^{k+1}, E^k)$ is indeed a $\C$-iso for all $i < n+m$. And by the long exact sequence of triples shown above we get a $\C$-iso $p_\ast : H_i(E^{k}, F) \to H_i(B^{k}, b_0)$ for all $i \leq n+m$. This finished the induction on $k$.
+	Now we conclude that $p_\ast : H_{i+1}(B^{k+1}, B^k) \to H_{i+1}(E^{k+1}, E^k)$ is indeed a $\C$-iso for all $i < n+m$. And by the long exact sequence of triples shown above we get a $\C$-iso $p_\ast : H_i(E^{k+1}, F) \to H_i(B^{k+1}, b_0)$ for all $i \leq n+m$. This finished the induction on $k$.
 
 	This concludes that $H_i(E, F) \to H_i(B, b_0)$ is a $\C$-iso and by another application of the long exact sequence (of the pair $(E,F)$) and the five lemma we get the $\C$-iso $H_i(E) \to H_i(B)$.
 }
@@ -81,7 +80,7 @@ In the following arguments we will consider fibrations and need to compute homol
 	Let $\C$ be a Serre class and $G \in \C$. Then for all $n > 0$ and all $i > 0$ we have $H_i(K(G, n)) \in \C$.
 }
 \Proof{
-	We prove this by induction on $n$. The base case $n = 1$ follows from group homology as the construction of $K(G, 1)$ can be used to obtain a projective resolution of $\Z$ as $\Z[G]$-module \todo{reference}. This then identifies the homology of the Eilenberg-MacLane space with the group homology, we get for $i>0$ an isomorphism
+	We prove this by induction on $n$. The base case $n = 1$ follows from group homology as the construction of $K(G, 1)$ can be used to obtain a projective resolution of $\Z$ as $\Z[G]$-module \todo{reference}. This then identifies the homology of the Eilenberg-MacLane space with the group homology which is in $\C$ by the axioms:
 	$$ H_i(K(G, 1); \Z) \iso H_i(G; \Z) \in \C. $$
 
 	Suppose we have proven the statement for $n$. If we consider the case of $n+1$ we can use the path fibration to relate it to the case of $n$:
@@ -89,7 +88,7 @@ In the following arguments we will consider fibrations and need to compute homol
 	Now $\Omega K(G, n+1) = K(G, n)$, and we can apply \LemmaRef{kreck} as the reduced homology of the fiber is in $\C$ by induction hypothesis. Conclude that the homology of $P K(G, n+1)$ is $\C$-isomorphic to the homology of $K(G, n)$. Since $\RH_\ast(P K(G, n+1)) = 0$, we get $\RH_\ast(K(G, n+1)) \in \C$.
 }
 
-For the main theorem we need the following construction. \todo{Geef de constructie of referentie}
+For the main theorem we need the following construction. \todo{referentie}
 \Lemma{whitehead-tower}{
 	(Whitehead tower)
 	We can decompose a $0$-connected space $X$ into fibrations:
@@ -168,8 +167,8 @@ For the main theorem we need the following construction. \todo{Geef de construct
 	\begin{enumerate}
 		\item $\pi_i(f)$ is a $\C$-iso for all $i < n$
 		\item $\pi_i(B_f, A) \in \C$ for all $i < n$
-		\item $\RH_i(B_f, A) \in \C$ for all $i < n$
+		\item $H_i(B_f, A) \in \C$ for all $i < n$
-		\item $\RH_i(f)$ is a $\C$-iso for all $i < n$.
+		\item $H_i(f)$ is a $\C$-iso for all $i < n$.
 	\end{enumerate}
 	Where (1) $\iff$ (2) and (3) $\iff$ (4) hold by exactness and (2) $\iff$ (3) by the Serre-Hurewicz theorem.
 }