Restructures the files

thesis/Makefile

@@ -20,16 +20,8 @@ haltfast: dirs
 	cd build; bibtex thesis
 	cp build/thesis.pdf ./
 
-symbols: dirs
-	xelatex -file-line-error -output-directory=build symbols.tex
-	xelatex -file-line-error -output-directory=build symbols.tex
-	scp build/symbols.pdf moerman@stitch.science.ru.nl:~/rathtpy_images.pdf
-	ssh moerman@stitch.science.ru.nl 'pdf2svg rathtpy_images.pdf rathtpy_images.svg'
-	scp moerman@stitch.science.ru.nl:~/rathtpy_images.svg ./symbols.svg
-
 dirs:
 	mkdir -p build
 	mkdir -p build/notes
 	mkdir -p build/diagrams
-	mkdir -p build/chapters
 	cp references.bib build/

thesis/chapters/Appendices.tex

@@ -1,4 +0,0 @@
-
-\input{notes/Algebra}
-\input{notes/Free_CDGA}
-\input{notes/Model_Categories}

thesis/chapters/Applications_And_Further_Topics.tex

@@ -1,4 +0,0 @@
-
-\input{notes/Calculations}
-
-\input{notes/Further_Topics}

thesis/chapters/Basics_Of_Rational_Homotopy_Theory.tex

@@ -1,4 +0,0 @@
-
-\input{notes/Basics}
-\input{notes/Serre}
-\input{notes/Rationalization}

thesis/notes/Algebra.tex

@@ -141,3 +141,55 @@ The first statement generalizes to a theorem where $A$ is a chain complex itself
 	$$ H(C) \tensor H(D) \tot{\iso} H(C \tensor D), $$
 	where we understand both tensors as graded. If $C$ and $D$ are algebras, this isomorphism is an isomorphism of algebras.
 \end{theorem}
+
+
+\section{The free cdga}
+\label{sec:free-cdga}
+
+Just as in ordinary linear algebra we can form an algebra from any graded module. Furthermore we will see that a differential induces a derivation.
+
+\begin{definition}
+	The \emph{tensor algebra} of a graded module $M$ is defined as
+	$$ T(M) = \bigoplus_{n\in\N} M^{\tensor n}, $$
+	where $M^{\tensor 0} = \k$. An element $m = m_1 \tensor \ldots \tensor m_n$ has a \emph{word length} of $n$ and its degree is $\deg{m} = \sum_{i=i}^n \deg{m_i}$. The multiplication is given by the tensor product (note that the bilinearity follows immediately).
+\end{definition}
+
+Note that this construction is functorial and it is free in the following sense.
+
+\begin{lemma}
+	Let $M$ be a graded module and $A$ a graded algebra.
+	\begin{itemize}
+		\item A graded map $f: M \to A$ of degree $0$ extends uniquely to an algebra map $\overline{f} : TM \to A$.
+		\item A differential $d: M \to M$ extends uniquely to a derivation $d: TM \to TM$.
+	\end{itemize}
+\end{lemma}
+
+\begin{corollary}
+	Let $U$ be the forgetful functor from graded algebras to graded modules, then $T$ and $U$ form an adjoint pair:
+	$$ T: \grMod{\k} \leftadj \grAlg{\k} :U $$
+	Moreover it extends and restricts to
+	$$ T: \dgMod{\k} \leftadj \dgAlg{\k} :U $$
+	$$ T: \CoCh{\k} \leftadj \DGA{\k} :U $$
+\end{corollary}
+
+As with the symmetric algebra and exterior algebra of a vector space, we can turn this graded tensor algebra in a commutative graded algebra.
+
+\begin{definition}
+	Let $A$ be a graded algebra and define
+	$$ I = \langle ab - (-1)^{\deg{a}\deg{b}}b a \I a,b \in A \rangle $$
+	Then $A / I$ is a commutative graded algebra.
+
+	For a graded module $M$ we define the \emph{free commutative graded algebra} as
+	$$ \Lambda(M) = TM / I $$
+\end{definition}
+
+Again this extends to differential graded modules (i.e. the ideal is preserved by the derivative) and restricts to cochain complexes.
+
+\begin{lemma}
+	We have the following adjunctions:
+	$$ \Lambda: \grMod{\k} \leftadj \grAlg{\k}^{comm} :U $$
+	$$ \Lambda: \dgMod{\k} \leftadj \dgAlg{\k}^{comm} :U $$
+	$$ \Lambda: \CoCh{\k} \leftadj \CDGA_\k :U $$
+\end{lemma}
+
+We can now easily construct cdga's by specifying generators and their differentials. Note that a free algebra has a natural augmentation, defined as $\counit(v) = 0$ for every generator $v$ and $\counit(1) = 1$.

thesis/notes/CDGA_Of_Polynomials.tex

@@ -1,79 +0,0 @@
-
-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:
-	$$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, d x_0, \ldots, d x_n)}{(\sum_{i=0}^n x_i - 1, \sum_{i=0}^n d x_i)}, $$
-	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. Also note that we have already seen the dual unit interval $\Lambda(t, dt)$ which is isomorphic to $\Apl_1$.
-
-These cdga's will assemble into a simplicial cdga when we define the face and degeneracy maps as follows ($j = 1, \ldots, n$):
-
-$$ d_i(x_j) = \begin{cases}
-	x_{j-1}, &\text{ if } i < j \\
-	0,       &\text{ if } i = j \\
-	x_j,     &\text{ if } i > j
-\end{cases} \qquad d_i : \Apl_n \to \Apl_{n-1} $$
-$$ s_i(x_j) = \begin{cases}
-	x_{j+1},       &\text{ if } i < j \\
-	x_j + x_{j+1}, &\text{ if } i = j \\
-	x_j,           &\text{ if } i > j	
-\end{cases} \qquad s_i : \Apl_n \to \Apl_{n+1} $$
-
-One can check that $\Apl \in \simplicial{\CDGA_\k}$. We will denote the subspace of homogeneous elements of degree $k$ as $\Apl^k$, this is a simplicial $\k$-module as the maps $d_i$ and $s_i$ are graded maps of degree $0$.
-
-\pagebreak
-\Lemma{apl-contractible}{
-	$\Apl^k$ is contractible.
-}
-\Proof{
-	We will prove this by defining an extra degeneracy $s: \Apl_n \to \Apl_{n+1}$. In the more geometric context of topological $n$-simplices we would achieve this by dividing by $1-x_0$. However, since this algebra consists of polynomials only, this cannot be done. Instead, we will multiply everything by $(1-x_0)^2$, so that we can divide by $1-x_0$. Define for $i = 1, \ldots, n$:
-	\begin{align*}
-		s(1) &= (1-x_0)^2 \\
-		s(x_i) &= (1-x_0) \cdot x_{i+1}
-	\end{align*}
-	Extend on the differentials and multiplicatively on $\Apl_n$. As $s(1) \neq 1$ this map is not an algebra map, however it well-defined as a map of cochain complexes. In particular when restricted to degree $k$ we get a linear map:
-	$$ s: \Apl^k_n \to \Apl^k_{n+1}. $$
-	Proving the necessary properties of an extra degeneracy is fairly easy. For $n \geq 1$ we get (on generators):
-	\begin{align*}
-		d_0 s(1)   &= d_0 (1 - x_0)^2 = (1 - 0) \cdot (1 - 0) = 1 \\
-		d_0 s(x_i) &= d_0((1-x_0)x_{i+1}) = d_0(1-x_0) \cdot x_i  \\
-		           &= (1-0) \cdot x_i = x_i
-	\end{align*}
-	So $d_0 s = \id$.
-	\begin{align*}
-		d_{i+1} s(1) &= d_{i+1} (1 - x_0)^2 = d_{i+1} (\sum_{j=1}^n x_j)^2 \\
-		             &= (\sum_{j=1}^{n-1} x_j)^2 = (1-x_0)^2 = s d_i(1)    \\
-		d_{i+1} s(x_j) &= d_{i+1}(1-x_0) d_{i+1}(x_j) = (1-x_0) d_i(x_{j+1}) = s d_i (x_j)
-	\end{align*}
-	So $d_{i+1} s = s d_i$. Similarly $s_{i+1} s = s s_i$. And finally for $n=0$ we have $d_1 s = 0$.
-
-	So we have an extra degeneracy $s: \Apl^k \to \Apl^k$, and hence (see for example \cite{goerss}) we have that $\Apl^k$ is contractible. As a consequence $\Apl^k \to \ast$ is a weak equivalence.
-}
-
-\Lemma{apl-kan-complex}{
-	$\Apl^k$ is a Kan complex.
-}
-\Proof{
-	By the simple fact that $\Apl^k$ is a simplicial group, it is a Kan complex \cite{goerss}.
-}
-
-Combining these two lemmas gives us the following.
-\Corollary{apl-extendable}{
-	$\Apl^k \to \ast$ is a trivial fibration in the standard model structure on $\sSet$.
-}
-
-Besides the simplicial structure of $\Apl$, there is also the structure of a cochain complex.
-\Lemma{apl-acyclic}{
-	$\Apl_n$ is acyclic, i.e. $H(\Apl_n) = \k \cdot [1]$.
-}
-\Proof{
-	This is clear for $\Apl_0 = \k \cdot 1$. For $\Apl_1$ we see that $\Apl_1 = \Lambda(x_1, d x_1) \iso \Lambda D(0)$, which we proved to be acyclic in the previous section.
-
-	For general $n$ we can identify $\Apl_n \iso \bigtensor_{i=1}^n \Lambda(x_i, d x_i)$, because $\Lambda$ is left adjoint and hence preserves coproducts. By the Künneth theorem \TheoremRef{kunneth} we conclude $H(\Apl_n) \iso \bigtensor_{i=1}^n H \Lambda(x_i, d x_i) \iso \bigtensor_{i=1}^n H \Lambda D(0) \iso \k \cdot [1]$.
-
-	So indeed $\Apl_n$ is acyclic for all $n$.
-}
-

thesis/notes/Calculations.tex

@@ -6,7 +6,7 @@ In this chapter we will calculate the rational homotopy groups of the spheres us
 
 \section{The sphere}
 
-\Proposition{}{
+\Proposition{odd-spheres-homotopy-groups}{
 	For odd $n$ the rational homotopy groups of $S^n$ are given by
 	$$ \pi_i(S^n) \tensor \Q \iso \begin{cases}
 		\Q, &\text{ if } i=n \\
@@ -26,7 +26,7 @@ In this chapter we will calculate the rational homotopy groups of the spheres us
 	$$ \pi_\ast(S^n) \tensor \Q = \pi_\ast(K(M_{S^n})) = \pi^\ast(M_{S^n})^\ast = \Q \cdot e^\ast. $$
 }
 
-\Proposition{}{
+\Proposition{even-spheres-homotopy-groups}{
 	For even $n$ the rational homotopy groups of $S^n$ are given by
 	$$ \pi_i(S^n) \tensor \Q \iso \begin{cases}
 		\Q, &\text{ if } i = 2n-1 \\
@@ -43,7 +43,7 @@ In this chapter we will calculate the rational homotopy groups of the spheres us
 
 The generators $e$ and $f$ in the last proof are related by the so called \Def{Whitehead product}. The whitehead product is a bilinear map $\pi_p(X) \times \pi_q(X) \to \pi_{p+q-1}(X)$ satisfying a graded commutativity relation and a graded Jacobi relation, see \cite{felix}. If we define a \Def{Whitehead algebra} to be a graded vector space with such a map satisfying these relations, we can summarize the above two propositions as follows \cite{berglund}.
 
-\Corollary{}{
+\Corollary{all-spheres-homotopy-groups}{
 	The rational homotopy groups of $S^n$ are given by
 	$$ \pi_\ast(S^n) \tensor \Q = \text{the free whitehead algebra on 1 generator}. $$
 }
@@ -55,7 +55,7 @@ Together with the fact that all groups $\pi_i(S^n)$ are finitely generated (this
 
 The following result is already used in proving the main theorem. But using the main theorem it is an easy and elegant consequence.
 
-\Proposition{}{
+\Proposition{em-space-homology-groups}{
 	For an Eilenberg-MacLane space of type $K(\Z, n)$ we have:
 	$$ H^\ast(K(\Z, n); \Q) \iso \Q[x], $$
 	i.e. the free graded commutative algebra on 1 generator.

thesis/notes/A_K_Quillen_Pair.tex → thesis/notes/Equivalence.tex

@@ -151,7 +151,7 @@ For the equivalence of rational spaces and cdga's we need that the unit and coun
 where the first of the two maps is given by the composition $X \to K(A(X)) \tot{K(m_X)} K(M(X))$,
 and the second map is obtained by the map $A \to A(K(A))$ and using the bijection from \CorollaryRef{minimal-model-bijection}: $[A, A(K(A))] \iso [A, M(K(A))]$. By the 2-out-of-3 property the map $A \to M(K(A))$ is a weak equivalence if and only if the ordinary unit $A \to A(K(A))$ is a weak equivalence.
 
-\Lemma{}{
+\Lemma{equivalence-base-case}{
 	(Base case) Let $A = (\Lambda(v), 0)$ be a minimal model with one generator of degree $\deg{v} = n \geq 1$. Then $A \we A(K(A))$.
 }
 \Proof{
@@ -162,7 +162,7 @@ and the second map is obtained by the map $A \to A(K(A))$ and using the bijectio
 	Now choose a cycle $z \in A(K(\Q^\ast, n))$ representing the class $x$ and define a map $A \to A(K(A))$ by sending the generator $v$ to $z$. This induces an isomorphism on cohomology. So $A$ is the minimal model for $A(K(A))$.
 }
 
-\Lemma{}{
+\Lemma{equivalence-pushout}{
 	(Induction step) Let $A$ be a cofibrant, connected algebra. Let $B$ be the pushout in the following square, where $m \geq 1$:
 	\begin{displaymath}
 		\xymatrix{
@@ -211,7 +211,7 @@ Note that by \RemarkRef{finited-dim-minimal-model} every cdga of finite type has
 
 Now we want to prove that $X \to K(M(X))$ is a weak equivalence for a simply connected rational space $X$ of finite type. For this, we will use that $A$ preserves and detects such weak equivalences by the Serre-Whitehead theorem (\CorollaryRef{rational-whitehead}). To be precise: for a simply connected rational space $X$ the map $X \to K(M(X))$ is a weak equivalence if and only if $A(K(M(X))) \to A(X)$ is a weak equivalence.
 
-\Lemma{}{
+\Lemma{topological-weak-equivalence}{
 	The map $X \to K(M(X))$ is a weak equivalence for $1$-connected, rational spaces $X$ of finite type.
 }
 \Proof{

thesis/notes/Free_CDGA.tex

@@ -1,51 +0,0 @@
-
-\section{The free cdga}
-\label{sec:free-cdga}
-
-Just as in ordinary linear algebra we can form an algebra from any graded module. Furthermore we will see that a differential induces a derivation.
-
-\begin{definition}
-	The \emph{tensor algebra} of a graded module $M$ is defined as
-	$$ T(M) = \bigoplus_{n\in\N} M^{\tensor n}, $$
-	where $M^{\tensor 0} = \k$. An element $m = m_1 \tensor \ldots \tensor m_n$ has a \emph{word length} of $n$ and its degree is $\deg{m} = \sum_{i=i}^n \deg{m_i}$. The multiplication is given by the tensor product (note that the bilinearity follows immediately).
-\end{definition}
-
-Note that this construction is functorial and it is free in the following sense.
-
-\begin{lemma}
-	Let $M$ be a graded module and $A$ a graded algebra.
-	\begin{itemize}
-		\item A graded map $f: M \to A$ of degree $0$ extends uniquely to an algebra map $\overline{f} : TM \to A$.
-		\item A differential $d: M \to M$ extends uniquely to a derivation $d: TM \to TM$.
-	\end{itemize}
-\end{lemma}
-
-\begin{corollary}
-	Let $U$ be the forgetful functor from graded algebras to graded modules, then $T$ and $U$ form an adjoint pair:
-	$$ T: \grMod{\k} \leftadj \grAlg{\k} :U $$
-	Moreover it extends and restricts to
-	$$ T: \dgMod{\k} \leftadj \dgAlg{\k} :U $$
-	$$ T: \CoCh{\k} \leftadj \DGA{\k} :U $$
-\end{corollary}
-
-As with the symmetric algebra and exterior algebra of a vector space, we can turn this graded tensor algebra in a commutative graded algebra.
-
-\begin{definition}
-	Let $A$ be a graded algebra and define
-	$$ I = \langle ab - (-1)^{\deg{a}\deg{b}}b a \I a,b \in A \rangle $$
-	Then $A / I$ is a commutative graded algebra.
-
-	For a graded module $M$ we define the \emph{free commutative graded algebra} as
-	$$ \Lambda(M) = TM / I $$
-\end{definition}
-
-Again this extends to differential graded modules (i.e. the ideal is preserved by the derivative) and restricts to cochain complexes.
-
-\begin{lemma}
-	We have the following adjunctions:
-	$$ \Lambda: \grMod{\k} \leftadj \grAlg{\k}^{comm} :U $$
-	$$ \Lambda: \dgMod{\k} \leftadj \dgAlg{\k}^{comm} :U $$
-	$$ \Lambda: \CoCh{\k} \leftadj \CDGA_\k :U $$
-\end{lemma}
-
-We can now easily construct cdga's by specifying generators and their differentials. Note that a free algebra has a natural augmentation, defined as $\counit(v) = 0$ for every generator $v$ and $\counit(1) = 1$.

thesis/notes/Homotopy_Relations_CDGA.tex

@@ -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 see as follows that it is a very good path object (\DefinitionRef{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 see as follows that it is a very good path object (\DefinitionRef{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.
 
 Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{right_homotopy}), however the converse need not be true.
 

thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex → thesis/notes/Homotopy_Theory_CDGA.tex

@@ -36,18 +36,3 @@ We furthermore have the following categorical properties of cdga's:
 
 \section{Homotopy groups of cdga's}
 \input{notes/Homotopy_Groups_CDGA}
-
-
-\Chapter{Polynomial Forms}{Adjunction}
-\label{sec:cdga-of-polynomials}
-
-\section{CDGA of Polynomials}
-\input{notes/CDGA_Of_Polynomials}
-
-\section{Polynomial Forms on a Space}
-\label{sec:polynomial-forms}
-\input{notes/Polynomial_Forms}
-
-
-\input{notes/Minimal_Models}
-\input{notes/A_K_Quillen_Pair}

thesis/chapters/Introduction.tex → thesis/notes/Introduction.tex

@@ -29,21 +29,3 @@ The main theorem is proven in \ChapterRef{Equivalence}. The adjunction from \Cha
 
 Finally we will see some explicit calculations in \ChapterRef{Calculations}. These calculations are remarkable easy. To prove for instance Serre's result on the rational homotopy groups of spheres, we construct a minimal model and read off their homotopy groups. We will also discuss related topics in \ChapterRef{Topics} which will conclude this thesis.
 
-
-\paragraph{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 also assumed, but the reader may review some facts on homological algebra in Appendix \ref{sec:algebra} and facts on model categories in Appendix \ref{sec:model_categories}.
-
-We will fix the following notations and categories.
-\begin{itemize}
-	\item $\k$ will denote a field of characteristic zero. Modules, tensor products, \dots\, are understood as $\k$-vector spaces, tensor products over $\k$, \dots.
-	\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 $\Top$: 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 $\Ab$: category of abelian groups and group homomorphisms.
-	\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 $|-| : \sSet \leftadj \Top : S$ to switch between topological spaces and simplicial sets.
-	\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).
-\end{itemize}
-
-\blankpage
-\tableofcontents
-\addcontentsline{toc}{section}{Contents}

thesis/notes/Minimal_Models.tex

@@ -148,7 +148,7 @@ The assignment to $X$ of its minimal model $M_X = (\Lambda V, d)$ can be extende
 	M_X \arwe[u]^{m_X} \ar[ur]^{f m_X} & M_Y \arwe[u]^{m_Y}
 	}
 \end{displaymath}
-Now by \LemmaRef{minimal-model-bijection} we get a bijection ${m_Y}_\ast^{-1} : [M_X, Y] \iso [M_X, M_Y]$. This gives a map $M(f) = {m_Y}_\ast^{-1} (f m_X)$ from $M_X$ to $M_Y$. Of course this does not define a functor of cdga's as it is only well defined on homotopy classes. However it is clear that it does define a functor on the homotopy category of cdga's.
+Now by \CorollaryRef{minimal-model-bijection} we get a bijection ${m_Y}_\ast^{-1} : [M_X, Y] \iso [M_X, M_Y]$. This gives a map $M(f) = {m_Y}_\ast^{-1} (f m_X)$ from $M_X$ to $M_Y$. Of course this does not define a functor of cdga's as it is only well defined on homotopy classes. However it is clear that it does define a functor on the homotopy category of cdga's.
 
 \Corollary{minimal-model-equivalence}{
 	The assignment $X \mapsto M_X$ defines a functor $M: \Ho(\CDGA_{\k,1}) \to \Ho(\CDGA_{\k,1})$. Moreover, since the minimal model is weakly equivalent, $M$ gives an equivalence of categories:

thesis/notes/Model_Categories.tex

@@ -178,7 +178,7 @@ Of course there is a completely dual definition of right homotopy, in terms of p
 	\end{itemize}
 }
 
-\Notation{cylinder_maps}{
+\Notation{path_maps}{
 	The map $p$ consists of two factors, which we will denote $p_0$ and $p_1$.
 }
 

thesis/notes/Polynomial_Forms.tex

@@ -1,4 +1,91 @@
 
+\Chapter{Polynomial Forms}{Adjunction}
+\label{sec:cdga-of-polynomials}
+
+\section{CDGA of Polynomials}
+
+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:
+	$$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, d x_0, \ldots, d x_n)}{(\sum_{i=0}^n x_i - 1, \sum_{i=0}^n d x_i)}, $$
+	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. Also note that we have already seen the dual unit interval $\Lambda(t, dt)$ which is isomorphic to $\Apl_1$.
+
+These cdga's will assemble into a simplicial cdga when we define the face and degeneracy maps as follows ($j = 1, \ldots, n$):
+
+$$ d_i(x_j) = \begin{cases}
+	x_{j-1}, &\text{ if } i < j \\
+	0,       &\text{ if } i = j \\
+	x_j,     &\text{ if } i > j
+\end{cases} \qquad d_i : \Apl_n \to \Apl_{n-1} $$
+$$ s_i(x_j) = \begin{cases}
+	x_{j+1},       &\text{ if } i < j \\
+	x_j + x_{j+1}, &\text{ if } i = j \\
+	x_j,           &\text{ if } i > j	
+\end{cases} \qquad s_i : \Apl_n \to \Apl_{n+1} $$
+
+One can check that $\Apl \in \simplicial{\CDGA_\k}$. We will denote the subspace of homogeneous elements of degree $k$ as $\Apl^k$, this is a simplicial $\k$-module as the maps $d_i$ and $s_i$ are graded maps of degree $0$.
+
+\pagebreak
+\Lemma{apl-contractible}{
+	$\Apl^k$ is contractible.
+}
+\Proof{
+	We will prove this by defining an extra degeneracy $s: \Apl_n \to \Apl_{n+1}$. In the more geometric context of topological $n$-simplices we would achieve this by dividing by $1-x_0$. However, since this algebra consists of polynomials only, this cannot be done. Instead, we will multiply everything by $(1-x_0)^2$, so that we can divide by $1-x_0$. Define for $i = 1, \ldots, n$:
+	\begin{align*}
+		s(1) &= (1-x_0)^2 \\
+		s(x_i) &= (1-x_0) \cdot x_{i+1}
+	\end{align*}
+	Extend on the differentials and multiplicatively on $\Apl_n$. As $s(1) \neq 1$ this map is not an algebra map, however it well-defined as a map of cochain complexes. In particular when restricted to degree $k$ we get a linear map:
+	$$ s: \Apl^k_n \to \Apl^k_{n+1}. $$
+	Proving the necessary properties of an extra degeneracy is fairly easy. For $n \geq 1$ we get (on generators):
+	\begin{align*}
+		d_0 s(1)   &= d_0 (1 - x_0)^2 = (1 - 0) \cdot (1 - 0) = 1 \\
+		d_0 s(x_i) &= d_0((1-x_0)x_{i+1}) = d_0(1-x_0) \cdot x_i  \\
+		           &= (1-0) \cdot x_i = x_i
+	\end{align*}
+	So $d_0 s = \id$.
+	\begin{align*}
+		d_{i+1} s(1) &= d_{i+1} (1 - x_0)^2 = d_{i+1} (\sum_{j=1}^n x_j)^2 \\
+		             &= (\sum_{j=1}^{n-1} x_j)^2 = (1-x_0)^2 = s d_i(1)    \\
+		d_{i+1} s(x_j) &= d_{i+1}(1-x_0) d_{i+1}(x_j) = (1-x_0) d_i(x_{j+1}) = s d_i (x_j)
+	\end{align*}
+	So $d_{i+1} s = s d_i$. Similarly $s_{i+1} s = s s_i$. And finally for $n=0$ we have $d_1 s = 0$.
+
+	So we have an extra degeneracy $s: \Apl^k \to \Apl^k$, and hence (see for example \cite{goerss}) we have that $\Apl^k$ is contractible. As a consequence $\Apl^k \to \ast$ is a weak equivalence.
+}
+
+\Lemma{apl-kan-complex}{
+	$\Apl^k$ is a Kan complex.
+}
+\Proof{
+	By the simple fact that $\Apl^k$ is a simplicial group, it is a Kan complex \cite{goerss}.
+}
+
+Combining these two lemmas gives us the following.
+\Corollary{apl-extendable}{
+	$\Apl^k \to \ast$ is a trivial fibration in the standard model structure on $\sSet$.
+}
+
+Besides the simplicial structure of $\Apl$, there is also the structure of a cochain complex.
+\Lemma{apl-acyclic}{
+	$\Apl_n$ is acyclic, i.e. $H(\Apl_n) = \k \cdot [1]$.
+}
+\Proof{
+	This is clear for $\Apl_0 = \k \cdot 1$. For $\Apl_1$ we see that $\Apl_1 = \Lambda(x_1, d x_1) \iso \Lambda D(0)$, which we proved to be acyclic in the previous section.
+
+	For general $n$ we can identify $\Apl_n \iso \bigtensor_{i=1}^n \Lambda(x_i, d x_i)$, because $\Lambda$ is left adjoint and hence preserves coproducts. By the Künneth theorem \TheoremRef{kunneth} we conclude $H(\Apl_n) \iso \bigtensor_{i=1}^n H \Lambda(x_i, d x_i) \iso \bigtensor_{i=1}^n H \Lambda D(0) \iso \k \cdot [1]$.
+
+	So indeed $\Apl_n$ is acyclic for all $n$.
+}
+
+
+\section{Polynomial Forms on a Space}
+\label{sec:polynomial-forms}
+
 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:

thesis/notes/Preliminaries.tex

@@ -0,0 +1,14 @@
+
+\paragraph{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 also assumed, but the reader may review some facts on homological algebra in Appendix \ref{sec:algebra} and facts on model categories in Appendix \ref{sec:model_categories}.
+
+We will fix the following notations and categories.
+\begin{itemize}
+	\item $\k$ will denote a field of characteristic zero. Modules, tensor products, \dots\, are understood as $\k$-vector spaces, tensor products over $\k$, \dots.
+	\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 $\Top$: 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 $\Ab$: category of abelian groups and group homomorphisms.
+	\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 $|-| : \sSet \leftadj \Top : S$ to switch between topological spaces and simplicial sets.
+	\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).
+\end{itemize}

thesis/notes/Rationalization.tex

@@ -100,7 +100,7 @@ Having rational cells we wish to replace the cells in a CW complex $X$ by the ra
 	with the obvious inclusion $\psi: X \to X_\Q$. By excision we see that $H_\ast(X_\Q, Y_\Q) \iso H_\ast(X \cup_f (Y \times I), Y \times {1}) = 0$. So by the long exact sequence of the inclusion we get $H_\ast(X_\Q) \iso H_\ast(Y_\Q)$, which proves by the rational Hurewicz theorem that $X_\Q$ is a rational space. At last we note that $H_\ast(X_\Q, X; \Q) \iso H_\ast(Y_\Q, Y; \Q) = 0$, since $\phi$ was a rationalization. This proves that $H_\ast(\psi; \Q)$ is an isomorphism, so by the rational Whitehead theorem, $\psi$ is a rationalization.
 }
 
-\Theorem{}{
+\Theorem{localization}{
 	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}
@@ -117,11 +117,11 @@ We will note prove that above theorem (it is analogue to \LemmaRef{SnQ-extension
 
 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{}{
+\Corollary{weak-eq-rationalization}{
 	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{}{
+\Corollary{rat-htpy-is-htpy-rat}{
 	The homotopy category of $1$-connected rational spaces is equivalent to the rational homotopy category of $1$-connected spaces.
 }
 

thesis/notes/Serre.tex

@@ -113,7 +113,7 @@ For the main theorem we need the following decomposition of spaces. The construc
 
 	It remains to show that $h_n$ is a $\C$-iso. Use the Whitehead tower from \LemmaRef{whitehead-tower} to obtain $\cdots \fib X(3) \fib X(2) = X$. Note that each $X(j)$ is $1$-connected and that $X(2) = X(1) = X$.
 
-	\Claim{}{For all $j < n$ and $i \leq n$ the induced map \linebreak $H_i(X(j+1)) \to H_i(X(j))$ is a $\C$-iso.}
+	\Claim{serre-step}{For all $j < n$ and $i \leq n$ the induced map \linebreak $H_i(X(j+1)) \to H_i(X(j))$ is a $\C$-iso.}
 	Note that $X(j+1) \fib X(j)$ is a fibration with $F = K(\pi_j(X), j-1)$ as its fiber. So by \LemmaRef{homology-em-space} we know $H_i(F) \in \C$ for all $i$. Apply \LemmaRef{kreck} to obtain a $\C$-iso $H_i(X(j+1)) \to H_i(X(j))$ for all $j < n$ and all $i > 0$. This proves the claim.
 
 	Considering this claim for all $j < n$ gives a chain of $\C$-isos $H_i(X(n)) \to H_i(X(n-1)) \to \cdot \to H_i(X(2)) = H_i(X)$ for all $i \leq n$. Consider the following diagram:

thesis/style.tex

@@ -1,9 +0,0 @@
-
-% lesser margins
-% \usepackage{geometry}
-% \geometry{a4paper}
-% \geometry{twoside=false}
-
-% no indent, but vertical spacing
-% \usepackage[parfill]{parskip}
-% \setlength{\marginparwidth}{2cm}

thesis/symbols.tex

@@ -1,20 +0,0 @@
-\documentclass[a4paper,12pt,footinclude=true,headinclude=true,oneside,dottedtoc]{scrbook}
-
-\usepackage{amsmath, amsthm}
-\usepackage[T1]{fontenc} 
-% use english, does not work with classicthesis
-% \usepackage{polyglossia}
-% \setmainlanguage[variant=british]{english}
-
-\usepackage[parts,drafting,eulerchapternumbers]{classicthesis}
-\setcounter{tocdepth}{0} % parts, chapters
-
-\input{preamble}
-
-\begin{document}
-% \pagenumbering{none}
-
-\[ f \quad \frac{1}{2}f \quad \frac{1}{6}f \quad S^1_\Q = \quad X \cdots \]
-\[ \xymatrix{ A \ar[rr] & & B } \]
-
-\end{document}

thesis/test_diagram.sh

@@ -1,8 +0,0 @@
-#!/bin/bash
-set -e
-
-file=$1
-sed "s|__INPUT__|$file|" test_diagram.tex | xelatex -file-line-error -output-directory=build
-mv build/texput.pdf test_diagram.pdf
-#exo-open test_diagram.pdf
-rm build/texput.*

thesis/test_diagram.tex

@@ -1,22 +0,0 @@
-\documentclass[a4paper, 12pt]{amsart}
-
-\input{style}
-\input{preamble}
-
-\begin{document}
-
-Your diagram:
-\cdiagrambase{__INPUT__}
-
-For reference, some default latex stuff:
-$$ A \cof B \we X \fib Y $$
-
-\newcommand{\newk}{\mathrm{I\!k}}
-
-$\newk$-linear, stuf over $\newk$
-
-\[ \k \newk \]
-
-\[ \CDGA_\k \CDGA_\newk \]
-
-\end{document}

thesis/thesis.tex

@@ -2,9 +2,6 @@
 
 \usepackage{amsmath, amsthm}
 \usepackage[T1]{fontenc} 
-% use english, does not work with classicthesis
-% \usepackage{polyglossia}
-% \setmainlanguage[variant=british]{english}
 
 \usepackage[parts,eulerchapternumbers]{classicthesis}
 \setcounter{tocdepth}{0} % parts, chapters
@@ -15,9 +12,10 @@
 \title{Rational Homotopy Theory}
 \author{Joshua Moerman}
 
+
 \begin{document}
-\pagenumbering{roman}
 
+\pagenumbering{roman}
 \begin{titlepage}
 \large
 
@@ -34,28 +32,46 @@
 	\vspace{7cm}
 
 	Radboud University Nijmegen\\
-	Supervisor: Ieke Moerdijk
+	Supervisor: Ieke Moerdijk\\
+	Second Reader: Javier J. Gutiérrez
 \end{center}
 
 \end{titlepage}
 \blankpage
 
-\include{chapters/Introduction}
+\input{notes/Introduction}
+\input{notes/Preliminaries}
+
+\blankpage
+\tableofcontents
+\addcontentsline{toc}{section}{Contents}
 \blankpage
 
+
 \pagenumbering{arabic}
 \part{Basics Of Rational Homotopy Theory}
-\include{chapters/Basics_Of_Rational_Homotopy_Theory}
+\input{notes/Basics}
+\input{notes/Serre}
+\input{notes/Rationalization}
+
 
 \part{CDGA's As Algebraic Models}
-\include{chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory}
+\input{notes/Homotopy_Theory_CDGA}
+\input{notes/Polynomial_Forms}
+\input{notes/Minimal_Models}
+\input{notes/Equivalence}
+
 
 \part{Applications and Further Topics}
-\include{chapters/Applications_And_Further_Topics}
+\input{notes/Calculations}
+\input{notes/Further_Topics}
+
 
 \appendix
 \part{Appendices}
-\include{chapters/Appendices}
+\input{notes/Algebra}
+\input{notes/Model_Categories}
+
 
 \bibliographystyle{alpha}
 \bibliography{references}