diff --git a/thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex b/thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex index 9df0bc2..8d22abb 100644 --- a/thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex +++ b/thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex @@ -1,15 +1,26 @@ \chapter{Homotopy Theory For cdga's} -Recall the following facts about cdga's over a ring $\k$: +Recall that a cdga $A$ is a commutative differential graded algebra, meaning that +\begin{itemize} + \item it has a grading: $A = \bigoplus_{n\in\N} A^n$, + \item it has a differential: $d: A \to A$ with $d^2 = 0$, + \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 eachother (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. + +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. + +We furthermore have the following categorical properties of cdga's: \begin{itemize} - \item A map $f: A \to B$ in $\CDGA_\k$ is a \emph{quasi isomorphism} if it induces isomorphisms in cohomology. \item The finite coproduct in $\CDGA_\k$ is the (graded) tensor product. \item The finite product in $\CDGA_\k$ is the cartesian product (with pointwise operations). \item The equalizer (resp. coequalizer) of $f$ and $g$ is given by the kernel (resp. cokernel) of $f - g$. Together with the (co)products this defines pullbacks and pushouts. \item $\k$ and $0$ are the initial and final object. \end{itemize} -In this chapter the ring $\k$ is assumed to be a field of characteristic zero. + +In this chapter the ring $\k$ is assumed to be a field of characteristic zero. In particular the modules are vector spaces. \section{Cochain models for the $n$-disk and $n$-sphere} \input{notes/CDGA_Basic_Examples} diff --git a/thesis/diagrams/Serre_Whitehead_LES.tex b/thesis/diagrams/Serre_Whitehead_LES.tex index 53d54bb..ac46cc6 100644 --- a/thesis/diagrams/Serre_Whitehead_LES.tex +++ b/thesis/diagrams/Serre_Whitehead_LES.tex @@ -1,4 +1,4 @@ -\xymatrix{ +\xymatrix @C=0.6cm{ \cdots \ar[r] & \pi_{i+1}(B_f, A) \ar[r] \ar[d] & \pi_i(A) \ar[r]^{f_\ast} \ar[d] & \pi_i(B) \ar[r] \ar[d] & \pi_i(B_f, A) \ar[r] \ar[d] & \cdots \\ \cdots \ar[r] & H_{i+1}(B_f, A) \ar[r] & H_i(A) \ar[r]^{f_\ast} & H_i(B) \ar[r] & H_i(B_f, A) \ar[r] & \cdots \\ } diff --git a/thesis/notes/Basics.tex b/thesis/notes/Basics.tex index ecec3f5..d59b9fd 100644 --- a/thesis/notes/Basics.tex +++ b/thesis/notes/Basics.tex @@ -16,7 +16,7 @@ In the following definition \emph{space} is to be understood as a topological sp $$ \pi_i(X) \tensor \Q \quad \forall i > 0.$$ } -In order to define the tensor product $\pi_1(X) \tensor \Q$ we need that the fundamental group is abelian, that is the rational homotopy groups are only defined for 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, 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. Note that for a rational space $X$, the ordinary homotopy groups are isomorphic to the rational homotopy groups, i.e. $\pi_i(X) \tensor \Q \iso \pi_i(X)$. diff --git a/thesis/notes/CDGA_Basic_Examples.tex b/thesis/notes/CDGA_Basic_Examples.tex index 3d455cb..e9ed406 100644 --- a/thesis/notes/CDGA_Basic_Examples.tex +++ b/thesis/notes/CDGA_Basic_Examples.tex @@ -1,6 +1,5 @@ -We will first define some basic cochain complexes which model the $n$-disk and $n$-sphere. $D(n)$ is the cochain complex generated by one element $b \in D(n)^n$ and its differential $c = d(b) \in D(n)^{n+1}$. \todo{Herschrijf} $S(n)$ is the cochain complex generated by one element $a \in S(n)^n$ with trivial differential (i.e. $d a = 0$). In other words: - +We will first define some basic cochain complexes which model the $n$-disk and $n$-sphere. $D(n)$ is the cochain complex generated by one element $b \in D(n)^n$ and its differential $c = d(b) \in D(n)^{n+1}$. On the other hand we define $S(n)$ to be the cochain complex generated by one element $a \in S(n)^n$ with trivial differential (i.e. $d a = 0$). In other words: $$ D(n) = ... \to 0 \to \k \to \k \to 0 \to ... $$ $$ S(n) = ... \to 0 \to \k \to 0 \to 0 \to ... $$ @@ -15,20 +14,20 @@ As we do not directly need this claim, we omit the proof. However, in the next s $S(n)$ plays a another special role: maps from $S(n)$ to some cochain complex $X$ correspond directly to elements in the kernel of $\restr{d}{X^n}$. Any such map is null-homotopic precisely when the corresponding elements in the kernel is a coboundary. So there is a natural isomorphism: $\Hom(S(n), X) / {\simeq} \iso H^n(X)$. -By using the free cdga functor we can turn these cochain complexes into cdga's $\Lambda(D(n))$ and $\Lambda(S(n))$. So $\Lambda(D(n))$ consists of linear combinations of $b^n$ \todo{gebruik niet weer $n$} and $c b^n$ when $n$ is even, and it consists of linear combinations of $c^n b$ and $c^n$ when $n$ is odd. In both cases we can compute the differentials using the Leibniz rule: -$$ d(b^n) = n \cdot c b^{n-1} $$ -$$ d(c b^n) = 0 $$ +By using the free cdga functor we can turn these cochain complexes into cdga's $\Lambda(D(n))$ and $\Lambda(S(n))$. So $\Lambda(D(n))$ consists of linear combinations of $b^k$ and $c b^k$ when $n$ is even, and it consists of linear combinations of $c^k b$ and $c^k$ when $n$ is odd. In both cases we can compute the differentials using the Leibniz rule: +$$ d(b^k) = k \cdot c b^{k-1} $$ +$$ d(c b^k) = 0 $$ -$$ d(c^n b) = c^{n+1} $$ -$$ d(c^n) = 0 $$ +$$ d(c^k b) = c^{k+1} $$ +$$ d(c^k) = 0 $$ Those cocycles are in fact coboundaries (using that $\k$ is a field of characteristic $0$): -$$ c b^n = \frac{1}{n} d(b^{n+1}) $$ -$$ c^n = d(b c^{n-1}) $$ +$$ c b^k = \frac{1}{k} d(b^{k+1}) $$ +$$ c^k = d(b c^{k-1}) $$ There are no additional cocycles in $\Lambda(D(n))$ besides the constants and $c$. So we conclude that $\Lambda(D(n))$ is acyclic as an algebra. In other words $\Lambda(j_n): \k \to \Lambda D(n)$ is a quasi isomorphism. -The situation for $\Lambda S(n)$ is easier: when $n$ is even it is given by polynomials in $a$, if $n$ is odd it is an exterior algebra \todo{?} (i.e. $a^2 = 0$). Again the sets $\Lambda(I) = \{ \Lambda(i_n) : \Lambda S(n+1) \to \Lambda D(n) \I n \in \N\}$ and $\Lambda(J) = \{ \Lambda(j_n) : \k \to \Lambda D(n) \I n \in \N\}$ play an important role. +The situation for $\Lambda S(n)$ is easier as it has only one generator (as algebra). For even $n$ this means it is given by polynomials in $a$. For odd $n$ it is an exterior algebra, meaning $a^2 = 0$. Again the sets $\Lambda(I) = \{ \Lambda(i_n) : \Lambda S(n+1) \to \Lambda D(n) \I n \in \N\}$ and $\Lambda(J) = \{ \Lambda(j_n) : \k \to \Lambda D(n) \I n \in \N\}$ play an important role. \begin{theorem} The sets $\Lambda(I)$ and $\Lambda(J)$ generate a model structure on $\CDGA_\k$ where: diff --git a/thesis/notes/Rationalization.tex b/thesis/notes/Rationalization.tex index be74c16..3b1f704 100644 --- a/thesis/notes/Rationalization.tex +++ b/thesis/notes/Rationalization.tex @@ -2,7 +2,7 @@ \chapter{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. +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. @@ -12,7 +12,6 @@ In this section we fix $n>0$. We will construct $S^n_\Q$ in stages $S^n(1), S^n( 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 $\pi_n(S^n) = \pi_n(S^n(r)) = \Z$. We will construct $S^n(r+1)$ with similar properties as follows. Let $f: S^n \to S^n(r)$ be a representative for $1 \in \Z \iso \pi_n(S^n(r))$ and $g: S^n \to S^n$ be a representative for $r+1 \in \Z \iso \pi_n(S^n)$. These maps combine into $\phi: S^n \to S^n \vee S^n \tot{f \vee g} S^n(r) \vee S^n$. We define $S^n(r+1)$ as the pushout in the following diagram. - \begin{displaymath} \xymatrix{ S^n \ar[r]^{\phi} \arcof[d] & S^n(r) \vee S^n \ar[d] \\ diff --git a/thesis/notes/Serre.tex b/thesis/notes/Serre.tex index a129587..fc2c474 100644 --- a/thesis/notes/Serre.tex +++ b/thesis/notes/Serre.tex @@ -158,7 +158,6 @@ For the main theorem we need the following construction. \todo{Geef de construct } \Proof{ Consider the mapping cylinder $B_f$ of $f$, i.e. factor the map $f$ as a cofibration followed by a trivial fibration $f: A \cof B_f \fib B$. The inclusion $A \subset B_f$ gives a long exact sequence of homotopy groups and homology groups: - \cdiagram{Serre_Whitehead_LES} We now have the equivalence of the following statements: